HULL, Circuit Judge:
Defendant John Williams appeals his conviction, as well as his sentence of 120 months’ imprisonment and subsequent 10 years’ supervised release. A jury found Williams guilty on one count of having used the internet to attempt to induce a minor to engage in sexual activity, in violation of 18 U.S.C. § 2422(b). On appeal, Williams argues that the district court erred in: (1) refusing to entertain a selective prosecution claim based on Williams’s socio-economic status; (2) precluding expert witness testimony concerning Williams’s susceptibility to inducement to commit this crime; and (3) refusing to modify the pattern jury instructions on reasonable doubt and entrapment at Williams’s request. After review of the record and the briefs, and with the benefit of oral argument, we affirm Williams’s conviction and sentence.
I. BACKGROUND
A. Undercover Law Enforcement Operations
From October 10, 2012, through October 15, 2012, the Volusia County, Florida Sheriffs Office conducted an undercover “sting” operation targeting individuals who were seeking, via the internet (“Craig-slist”), to have sexual relations with minors. The operation led to the arrest of twenty-three individuals including Williams, a sixty-six-year-old cardiac anesthesiologist.
B. Offense Conduct
On October 11, 2012, Tallahassee Police Sergeant Sonia Bush, who was working as part of the Volusia County Sheriffs Office investigation, posted an advertisement in the “personals,” “casual encounters” section of Craigslist entitled, “still need help, need teacher-w4m-43 (Ormond beach).” “W4m” stood for “Women for men.” The body of the ad read, “Been trying to find the right discreet man to teach my daughter.”
Twenty minutes later, Williams responded to the ad, saying, “[W]hat age group are you interested in? ? 56 yo nice looking professional, very gentle and romantic.” He identified himself as “John.” Sergeant Bush responded, saying, “[N]ot really looking for age range,,, looking for experience, gentleness, and patience,, and of course someone who is good at teaching a 14-year-old young lady.” Twenty minutes after that, Williams responded, saying, “Not for me sorry[.] John cell (803) 429 4924.”
Sergeant Bush then asked Williams why he gave her his number if he was not interested. Williams responded:
Well when you told me her age I was afraid of a legal situation.
How would I know this is not a trap? ? can we exchange pics? ? or can you give me more info/description? ?
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meet for brunch and talk this over ?'?
Sergeant Bush responded by sending a picture of a young girl on a swing, explaining that “Sydney” was 14 years old and
petite, with long dark hair and hazel eyes. Sergeant Bush said that she did not know how to ease his fears that he was being trapped but that, although she was not interested in men, her daughter was.
The next morning, on October 12, 2012, Williams responded by sending a picture of himself and saying, “I am sending my pic, see what she thinks[.]” When Sergeant Bush did not respond, Williams contacted her again a few hours later, saying, “[G]uess you found someone to meet your needs ? ?” and, “Nicole, I am guessing you found a better match for your needs but if you havent I hope you will still consider me[.] John.”
At 6:30 p.m. that evening, Sergeant Bush replied, telling Williams that she still had not found anyone. Two minutes later, Williams asked her if she had received his picture and told her that he was in the Port Orange, Florida area.- Williams then asked again whether she had received his picture and told her that he was in Port Orange. Williams asked her if they would like to meet for something to eat or drink and said, “[w]e could talk it over and check each other out.” Sergeant Bush told him that she had received- his picture and would show it to her daughter when she got home. Williams told her that he would wait for her reply.
Later that day, Sergeant Bush told Williams that she was “in the process of screening different men” and that “the biggest thing I need to know is what you would teach her.” In response, Williams sent Sergeant Bush a long message, describing in detail how he planned to perform a variety of sexual acts with the girl.
At that point, Sergeant Bush showed the emails to her supervisors, who determined that probable cause existed that a crime had been committed. Sergeant Bush then responded to Williams, saying that everything sounded wonderful except that her daughter wanted “her first time to be in her bedroom.” Williams replied that would be fine but that they had to meet first. Sergeant Bush suggested that Williams come over to her house where her daughter could make dinner. After exchanging more messages, Williams agreed to come to the mother’s house that night. Sergeant Bush provided Williams with directions to her home.
That night, Williams traveled to the address that Sergreant Bush gave him. Once Williams arrived, he got out of his car and approached the front door carrying a towel, a plastic grocery bag, and a canvas bag. Law enforcement agents arrested Williams as he opened the door.
• Williams’s grocery bag contained two bottles of wine, condoms, surgical scrub pants, a man’s bathrobe, two bottles of lubricant, several sex toys, shaving cream and a razor, toothpaste, a toothbrush, and two hairbrushes. A subsequent search of Williams’s vehicle revealed a large variety of additional sex toys and devices, as well as ointments and lubricants. Williams had far more paraphernalia than necessary to commit this single crime.
Further, following his arrest, Williams told a federal agent that he owned his own plane and had flown from his home in South Carolina to Florida on October 10, 2012, the day before he began exchanging emails with Sergeant Bush.
C. Williams’s State Court Case and Transfer
Following Williams’s arrest, Williams was charged in state court with: (1) use of a computer to seduce, solicit, or lure a child, in violation of Fla. Stat. § 847.0135(3)(a); (2) traveling to meet a minor after use of a computer to solicit a child, in violation of Fla. Stat. § 847.0135(4)(a); and (3) unlawful use of a
two way communication device, in violation of Fla. Stat. § 934.215.5. All of the twenty-two other arrestees also arrested during the Volusia County Sheriffs Office’s October 10-October 15, 2015 undercover operation were charged with the same three state law crimes.
In February 2013, the Assistant State Attorney (“ASA”) assigned to Williams’s case, Tiffany Adleman, received a telephone call from Immigration and Custom Enforcement Special Agent Joe Grey. Grey requested that Williams’s state case be dismissed in favor of a federal prosecution. ASA Adleman offered to transfer all of the cases that were filed in connection with the Volusia County Sheriffs Office undercover operation. However, Agent Grey declined and was only interested in prosecuting Williams.
D.Procedural History in Federal Court
On February 6, 2013, a federal grand jury returned an indictment charging Williams with one count of using the internet to knowingly attempt to persuade, induce, and entice an individual who had not attained the age of 18 years to engage in sexual activity for which any person could be charged with a criminal offense under Florida law, namely, lewd or lascivious battery, a violation of Florida Statute 800.04, all in violation 18 U.S.C. § 2422(b). The count carried a mandatory minimum penalty of ten years’ imprisonment.
On March 19, 2013, the State of Florida dismissed the three state charges pending against Williams in favor of the federal prosecution.
E. Williams’s Psychiatric Examination
On April 5, 2013, prior to Williams’s federal trial, Williams filed with the district court an unopposed motion for psychological, psychiatric, neuropsychological, and neurological evaluations, pursuant to 18 U.S.G. § 4241(b), and to determine competency. The district court referred the motion to a magistrate judge, who granted the motion and ordered Williams to undergo a psychiatric examination to determine whether he was competent to stand trial. Williams was transferred to a federal detention center in Miami, Florida for a psychiatric examination. On April 15, 2013, the psychiatric examination began.
F. Williams’s First Selective Prosecution Motion
On September 30,2013, while the results of Williams’s psychiatric examination were pending, Williams filed a motion to compel discovery and/or to dismiss the indictment due to selective prosecution on the basis of his age,
wealth, and socio-economic status. In the motion, Williams noted that, although he had been one of the twenty-three people who were arrested during the Volusia County Sheriffs Office undercover operation, only his case was selected for federal prosecution. Williams requested that the district court allow him discovery to determine whether he had been unconstitutionally selected for federal prosecution.
The motion was referred to a magistrate judge. On December 10, 2013, the magistrate judge issued a report and recommendation finding that Williams had successfully demonstrated that he was similarly situated to the arrestees who were not prosecuted in federal court. The magis
trate judge recommended, however, that Williams’s motion be denied because he failed to show that the government’s prosecution was based on a constitutionally impermissible motive. On April 21, 2014, the district court adopted the report and recommendation and denied Williams’s motion.
G.Williams’s First Competency Hearing Leads to a Determination that Williams Is Not Competent to Stand Trial
On May 13 and 14, 2014, nearly a year after Williams’s transfer to Miami for a psychiatric examination, a magistrate judge conducted a competency hearing regarding the results of the examination. At the hearing, the magistrate judge heard testimony from: (1) Dr. Rodolfo Buigas, a forensic psychologist from the Bureau of Prisons who had examined an MRI of Williams’s brain; (2) Dr. Kenneth C. Fischer, who had conducted a neurological evaluation of Williams; (3) Dr. Frank L. Quinn, a counselor who had worked with both Williams and his wife over several years; and (4) Dr. Michael Rappaport, a clinical psychologist who had evaluated Williams’s competency post-arrest. Following their testimony, the magistrate judge recommended that the district court find that Williams was not competent to stand trial. Neither party objected to the recommendation. On June 2, 2014, the district court adopted the recommendation and referred Williams to the Federal Medical Center at Butner, North Carolina for further evaluation and “restoration” treatment.
On August 19, 2014, Williams was transferred to the Federal Medical Center at Butner, North Carolina. Dr. Maureen Reardon, a forensic psychologist, and Dr. Tracy O’Connor Pennuto, Butner’s staff neuropsychologist, completed a neuropsy-chological consultation of Williams. On January 5, 2015, following months of treatment, Drs. Reardon and Pennuto completed a written evaluation of Williams and emailed it to the district court. The written evaluation declared that Williams was now competent to stand trial. Upon receiving the written evaluation, a magistrate judge scheduled Williams for a second competency hearing, to be held on May 20, 2015.
H. Government Statement Prompting Renewed Selective Prosecution Motion
On or about April 7, 2015, prior to Williams’s second competency hearing, Williams’s counsel met with the government to discuss a number of other pretrial matters. At this meeting, Williams’s counsel expressed frustration about Williams’s alleged selective prosecution. In response, an Assistant United States Attorney (“AUSA”) responded: “How many [of the co-arrestees] were doctors who flew their own planes?”
On April 23, 2015, Williams’s counsel notified the district court that, partially as a result of the AUSA’s remark, Williams would be filing a renewed motion on Williams’s selective prosecution claim. On July 13, 2015, Williams filed a renewed motion to dismiss the indictment and/or for limited discovery due to selective prosecution. In the motion, Williams argued that he had been selectively chosen for prosecution on account of his wealth and socioeconomic status. The district court denied Williams’s renewed selective prosecution motion, finding that Williams again failed to establish that the government’s prosecution was due to a discriminatory motive.
I. Williams’s Second Competency Hearing Leads to a Determination that Williams Is Competent to Stand Trial
On May 20, 2015, a magistrate judge conducted Williams’s second competency
hearing. During the hearing, both Dr. Reardon and Dr. Pennuto testified regarding their examination of Williams and their report, in which they found that Williams was competent to stand trial. Dr. Pennuto testified that she had determined that most of Williams’s cognitive functioning was fully intact. Dr. Reardon testified that, although Williams had suffered a mild decline in some areas of cognitive abilities, he was still functioning very well and higher than most.
The magistrate judge entered a sealed report and recommendation, recommending that the district court find that Williams was competent to stand trial. On July 13, 2015, the district court issued an order adopting the recommendation and setting the case for trial, to be held on August 26, 2015.
J. Government’s Motions In Limine; Daubert Hearing
On August 10, 2015, the United States filed a motion in limine seeking to exclude Dr. Rappaport’s testimony from the first competency hearing concerning Williams’s mental condition. Williams filed a response, asserting that Dr. Rappaport’s opinion was admissible as to his entrapment defense. Williams also gave notice that he intended to introduce at trial the testimony provided by Drs. Fischer and Quinn at his first competency hearing. The government filed an additional motion in limine seeking to exclude their testimony as well.
On August 24, 2015, the district court entered an order stating that it would need to address the various motions in limine through a Daubert hearing to determine the admissibility of the proffered experts.
At the beginning of trial on August 26, 2015, the district court conducted the Dau-bert hearing.
At the hearing, Dr. Rappaport testified that, in March 2013, he .conducted a mental status examination of Williams and referred him to Dr. Fischer for neurological testing to determine the cause of Williams’s purported neurological deficiencies.
Dr. Rappaport testified that Dr. Fischer’s resulting neurological report—showing that Williams had some form of brain disease—corroborated Dr. Rappaport’s suspicions about Williams’s cognitive deficiencies. Dr. Rappaport concluded that Williams’s “executive function was impaired and that he was very susceptible to certain things.” Dr. Rappaport testified further that he had determined that Williams had “executive-functioning problems,” i.e., problems with his decision-making process.
Williams’s counsel asked Dr. Rappaport to explain his basis for concluding from his March 2013 examination that Williams was “unduly susceptible to certain things.” Dr. Rappaport responded that Williams was a “68-year-old brain-impaired man, who was at that time by himself, isolated and very lonely, ,.. who could easily have been manipulated by other people much more easily manipulated than [his attorney] or you or me or the court reporter.” The district court asked: “Based upon what peer-reviewed study?” Dr. Rappaport replied: “Based upon my examination of him ... and the validation of my examination by [Dr. Fischer] .., who evaluated [the]
actual picture of his brain.” Dr. Rappaport also testified that “[t]here’s all kinds of stuff about how older people are susceptible to manipulation.”
On cross-examination, Dr. Rappaport confirmed that he had first seen Williams in March 2013, six months after his arrest. Dr. Rappaport further testified that none of the tests that he had administered in March 2013 were capable of assessing Williams’s mental condition on the night of his arrest in October 2012. However, Dr. Rappaport testified that he could assess the condition of Williams’s mental state on that night by considering Williams’s later behavior, as well as “what [Williams’s] family had told [him].”
Government counsel asked Dr. Rappa-port to explain how he could show that Williams’s mental state made him susceptible to inducement to commit this crime in October 2012. Dr. Rappaport responded by saying that “the best predictor usually of future behavior is past behavior” and that Dr. Fischer’s neurological conclusion—that the changes in Williams’s brain had been occurring for some time—was consistent with what he had learned about Williams’s behavior from Williams’s family members.
Following the hearing, the district court granted the government’s motions in li-mine, finding that none of Williams’s doctors’ testimony met the Daubert standard for admissibility.
As to Drs. Fischer and Quinn, the district court first noted that Dr. Fischer had not attempted to connect his findings on the physical changes to Williams’s brain to his particular susceptibility to inducement to commit this crime—he had opined only on Williams’s competency to stand trial. The district court further noted that Dr. Quinn had not correlated his observations of Williams to Williams’s susceptibility to inducement. Therefore, the district court determined that the reports and testimony of Drs. Fischer and Quinn would only be admissible to the extent that they provided foundational evidence in support of Dr. Rappaport’s testimony, the only doctor to testify on the relationship between Williams’s mental state and his susceptibility to inducement to commit this crime.
As to Dr. Rappaport, the district court noted that he had been unable to identify any peer-reviewed methodology for identifying how, or to what extent, a person’s cognitive decline causes him to be more susceptible to inducement. The district court stated:
There’s no indication that Dr. Rappa-port’s analysis in terms of the connection between what he observed and what Dr. Fischer saw and the ultimate conclusion of susceptibility is in fact subject to peer review publication or has been or can—that it can be repeated in a reliable manner. There is no indication in the record that I saw that the technique of a clinical psychologist evaluating a patient and relying upon the objective findings of the neurologist is generally accepted in the scientific community to demonstrate susceptibility to suggestion.
The district court further noted that Dr. Rappaport could not point to a scientific methodology showing how his March 2013 examination could accurately show that Williams was susceptible to inducement to commit this crime in October 2012.
Accordingly, the district court determined that Dr. Rappaport’s testimony was not admissible under the Daubert standard. As a result, the district court further determined that the admission of the testimony of Drs. Quinn and Fischer would be more prejudicial than probative since neither had opined on susceptibility to inducement and them testimony could only be foundational to Dr. Rappaport’s now inadmissible testimony.
K. Defense’s Proffered Testimony
Following the district court’s ruling on the admissibility of the testimony of Drs. Rappaport, Fischer, and Quinn, Williams’s counsel proffered the testimony that the doctors would have presented if the district court had not granted the government’s motions in limine.
Williams’s counsel proffered that, had Dr. Quinn been permitted to testify, he would have stated that by July 8, 2002— approximately ten years prior to Williams’s arrest—Dr. Quinn had diagnosed Williams with a major mental disease or defect. Dr. Quinn would have explained that, on October 24, 2012, shortly after Williams’s arrest, Williams requested that Dr. Quinn conduct a mental status evaluation of Williams because Williams was concerned about his cognitive function and poor memory. Shortly after Williams’s October 2012 arrest, Dr. Quinn would have diagnosed Williams with recurrent attention deficit disorder and depression. Dr. Quinn would have testified that, during the time leading up to Williams’s arrest, Williams was in a “fugue” state or a manic episode brought on by withdrawal from prescription medication.
In addition, Williams’s counsel proffered that, had Dr. Fischer testified, he would have stated that, since 1971, he frequently appeared as an expert witness in federal and state court proceedings as a treating neurologist. He would have testified that he conducted a comprehensive neurological evaluation of Williams in May 2013 while Williams was in custody at the federal detention center in Miami. The evaluation included review and assessment of Williams’s medical history and executive functioning, which Dr. Fischer described as Williams’s ability to organize his life and carry out specific activities in a coherent manner. Dr. Fischer would have testified that his evaluation concluded that Williams was suffering from some form of dementia and cognitive decline, which necessarily adversely effected Williams’s executive functioning, i.e,, decision-making ability.
Williams’s counsel proffered that, had Dr. Rappaport testified at trial, he would have testified that he initially met with Williams on March 11,- 2013, in the Seminole County Jail and examined him. Dr. Rappaport would have testified that, in 2013, Williams: (1) presented with a “flat” affect; (2) was unable to count backwards from 100 by 7; and
(3) showed clearly poor problem-solving skills. Dr. Rappaport would have testified to a “reasonable psychological certainty” that someone suffering from the same or similar mental health deficits and defects as Williams would have been more susceptible to suggestion, manipulation, and inducement to commit this crime by third parties.
L. . Reasonable Doubt Jury Instruction
On August 28, 2015, the third day of trial, the district court held a jury-instruction conference. At the jury-instruction conference, Williams objected to the district court’s proposed use of the Eleventh Circuit pattern jury instruction on reason
able doubt. The pattern jury instruction reads, in relevant part: “The Government’s burden of proof is heavy, but it doesn’t have to prove a Defendant’s guilt beyond all possible doubt. The Government’s proof only has to exclude any ‘reasonable doubt’ concerning the Defendant’s guilt.”
Through his objection, Williams sought to strike the word “only” from the pattern jury instruction, arguing that it lessened the government’s burden of proof. The district court overruled the objection, explaining: “I think the instruction in its totality makes it clear that the jury doesn’t have to exclude all possible doubt, including hypothetical or fanciful doubt, but only a reasonable doubt, which is then [later] defined.” The district court ultimately read to the jury the full pattern jury instruction on reasonable doubt, including the word “only.”
M. Entrapment Jury Instruction
Williams also requested that the district court modify the Eleventh Circuit pattern jury instruction concerning entrapment. Williams proposed to add three provisions (underlined below) to the relevant portion of the pattern jury instruction on entrapment:
But there is no entrapment when a defendant is willing to violate the law and the Government merely provides what appears to be a favorable opportunity for the defendant to commit a crime. However, in determining one’s willingness or unwillingness to violate the law, the jury is permitted to consider evidence of the defendant’s state of mind and diminished mental health.
For example, it’s not entrapment for a Government agent to pretend to be someone else and offer—directly or through another person—to engage in an unlawful transaction. However, it is entrapment if the defendant, as a direct result of diminished mental health, was more easily induced and/or persuaded by the Government to engage in the unlawful behavior.
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But if, as a result of the defendant’s diminished mental health, there is a reasonable doubt about whether the Defendant was willing to commit the crime without the persuasion of a Government officer or a person under the Government’s direction, then you must find the Defendant not guilty.
The government did not object to the inclusion of Williams’s first proposed revision but did object to the inclusion of the others. As to the second proposed revision, the district court determined that revision should not be included because, without the testimony of Dr. Rappaport, no witness testimony supported application of the proposition to this case. As to the third proposed revision, the district court stated that, because “diminished mental health, proven through the proper means or matter, can and should be considered in terms of inducement,” the third proposed revision was appropriate. The district court agreed to include the first and third proposed revisions but not the second proposed revision.
However, after Williams presented his defense case, the district court told counsel for both parties that it would “give the Eleventh Circuit pattern [on entrapment] without a reference to mental disease or defect ... because ... mental disease or defect requires evidence from a licensed physician of some kind that can draw the nexus.” During the jury instructions, the district court read the pattern jury instruction on entrapment to the jury without including Williams’s first and third proposed revisions.
On August 31, 2015, the jury returned a verdict of guilty against Williams on the charged offense. On November 24, 2015, Williams timely appealed.
II. SELECTIVE PROSECUTION
A. Standard of Review
In reviewing the denial of a motion to dismiss for selective prosecution, this Court reviews the district court’s factual findings for clear error and its legal conclusions de novo. United States v. Jordan, 635 F.3d 1181, 1185 (11th Cir. 2011). This Court reviews for abuse of discretion a district court’s denial of a request for discovery in a selective prosecution claim. Id. “An abuse of discretion can occur where the district court applies the wrong law, follows the wrong procedure, bases its decision on clearly erroneous facts, or commits a clear error in judgment.” United States v. Brown, 415 F.3d 1257, 1266 (11th Cir. 2005).
B. Relevant Law
Selective prosecution claims are analyzed under the equal protection component of the Fifth Amendment’s Due Process Clause. See United States v. Armstrong, 517 U.S. 456, 464, 116 S.Ct. 1480, 1486, 134 L.Ed.2d 687 (1996). Equal protection dictates that “the decision whether to prosecute may not be based on an unjustifiable standard such as race, religion, or other arbitrary classification.” Jordan, 635 F.3d at 1188 (internal citations and quotations omitted).
“In order to dispel the presumption that a prosecutor has not violated equal protection, a criminal defendant must present clear evidence to the contrary.” United States v. Smith, 231 F.3d 800, 807 (11th Cir. 2000) (quoting Armstrong, 517 U.S. at 465, 116 S.Ct. at 1486). To establish a selective prosecution claim, the defendant must show that: (1) the “prosecution had a discriminatory effect, i.e., that similarly situated individuals were not prosecuted,” and; (2) “that the difference in treatment, or selectivity of the prosecution, was motivated by a discriminatory purpose.” Id. at 809. The discriminatory purpose prong requires that “the decisionmaker selected or reaffirmed a particular course of action at least in part ‘because of,’ not merely ‘in spite of,’ its adverse effects upon an identifiable group.” Jordan, 635 F.3d at 1188 (quoting Wayte v. United States, 470 U.S. 598, 610, 105 S.Ct. 1524, 1532, 84 L.Ed.2d 547 (1985)).
A defendant may obtain discovery in support of a selective prosecution claim where the defendant provides “some evidence tending to show the existence of the essential elements of the defense.” Armstrong, 517 U.S. at 468, 116 S.Ct. at 1488.
C.Discussion
' The government does not dispute the district court’s determination that Williams was similarly situated to the other arres-tees in that Williams committed the same crime in similar circumstances.
Accordingly, we limit our review to whether the government selectively prosecuted Williams based on a discriminatory motive or, alternatively, whether the district court abused its discretion in denying Williams the opportunity to conduct discovery on that question.
In his brief on appeal, Williams recognizes that “there does not (yet) appear to be a Federal case dismissing an indictment
due to selective prosecution on the basis of a defendant’s wealth.” Nonetheless, Williams asserts that the “law is clear” that “wealth and socioeconomic status are ‘off limits’ as far as disparate treatment in the criminal justice system is concerned.” See McDonald v. Bd. of Election Comm’rs, 394 U.S. 802, 807, 89 S.Ct. 1404, 1407-08, 22 L.Ed.2d 739 (1969) (“[A] careful examination on our part is especially warranted where lines are drawn on the basis of wealth or race ... two factors which would independently render a classification highly suspect and thereby demand a more exacting judicial scrutiny.”); Harper v. Va. Bd. of Elections, 383 U.S. 663, 668, 86 S.Ct. 1079, 1082, 16 L.Ed.2d 169 (1966) (“Wealth, like race, creed, or color, is not germane to one’s ability to participate intelligently in the electoral process.”); Griffin v. Illinois, 351 U.S. 12, 19, 76 S.Ct. 585, 591, 100 L.Ed. 891 (1956) (“There can be no equal justice where the kind of trial a man gets depends on the amount of money he has.”).
Even assuming arguendo that Williams’s argument—that prosecution may not be based on one’s wealth vis-á-vis other potential defendants—is true, Williams points to no credible evidence suggesting that the government selectively chose to prosecute him because of his wealth.
Williams’s alleged evidence offered to show discriminatory intent is the AUSA’s comment to Williams’s counsel—“How many [of the co-arrestees] were doctors who flew their own planes?”—made over two years after Williams’s federal indictment. This comment does not specifically mention wealth at all. And though Williams asks us to infer a concern about Williams’s wealth from the comment, under the factual circumstances of this case, the comment more reasonably reflects the prosecutor’s concern about Williams’s mobility to readily travel to where his minor victim was or might be once Williams made online contact, or to spirit the child away. In fact, Williams flew the day before from South Carolina to Florida, which enabled him to go to the victim’s residence the next day. Williams also provides no concrete evidence as to the relative wealth or poverty of the other arrestees.
Without more, Williams has provided no credible evidence showing that the government harbored a discriminatory motive against Williams on the basis of his wealth or status. Accordingly, the district court did not err in denying Williams’s motion to dismiss the indictment on this basis or in refusing to allow Williams to conduct discovery on the issue.
III. ADMISSIBILITY OF EXPERT TESTIMONY
This Court reviews “district eourt[ ] decisions regarding the admissibility of expert testimony and the reliability of an expert opinion” for abuse of discretion. United States v. Frazier, 387 F.3d 1244, 1258 (11th Cir. 2004).
B. Applicable Law
A successful entrapment defense has two elements: (1) governmental inducement of the crime; and (2) lack of predisposition on the part of the defendant. United States v. Brown, 43 F.3d 618, 623 (11th Cir. 1995). The defendant bears the initial burden of production to show government inducement. United States v. Sistrunk, 622 F.3d 1328, 1333 (11th Cir. 2010). Once the defendant has met his burden to show evidence of government inducement, the burden shifts to the government to prove beyond a reasonable doubt that the defendant was predisposed to commit the crime, and the question of entrapment becomes a
factual one for the jury to decide. United States v. Ryan, 289 F.3d 1389, 1344 (11th Cir. 2002).
A defendant may show inducement by producing evidence sufficient to raise a jury issue “that the government’s conduct created a substantial risk that the offense would be committed by a person other than one ready to commit it.” Brown, 43 F.3d at 623 (quoting United States v. Andrews, 765 F.2d 1491, 1499 (11th Cir. 1985)). Evidence of the government’s “mere suggestion of a crime or initiation of contact is not enough.” Id. Rather, “government inducement requires an element of persuasion or mild coercion,” which consists of “excessive pressure or manipulation of a non-criminal motive.” Id.
“Regardless of the defendant’s ability to engage in criminal acts ... the prompt commission of the crime at the first opportunity is enough to show predisposition.” Id. at 624.
Williams sought to introduce the testimony of Drs. Rappaport, Fischer, and Quinn as expert testimony. An expert’s testimony is admissible if: (1) the expert’s “specialized knowledge will help the trier of fact to understand the evidence or to determine a fact in issue”; (2) “the testimony is based on sufficient facts or data”; (3) “the testimony is the product of reliable principles and methods”; and (4) “the expert has reliably applied the principles and methods to the facts of the case.” Fed. R. Evid. 702. “The task of ensuring that an expert’s testimony both rests on a reháble foundation and is relevant to the task at hand is assigned to the district court.” United Fire & Gas. Co. v. Whirlpool Corp., 704 F.3d 1338, 1341 (11th Cir. 2013) (quoting Daubert v. Merrell Dow Pharm., Inc., 509 U.S. 579, 597, 113 S.Ct. 2786, 2799, 125 L.Ed.2d 469 (1993)). “The admissibility standard is a liberal one, and ... the rejection of expert testimony is the exception rather than the rule.” Frazier, 387 F.3d at 1293-94.
In Daubert v. Merrell Dow Pharmaceuticals, Inc., the Supreme Qourt established several factors for courts to consider in determining whether to admit an expert’s testimony: (1) whether the expert’s methodology has been tested or is capable of being tested; (2) whether the theory or technique used by the expert has been subjected to peer review and publication; (3) whether there is a known or potential error rate of the methodology; and (4) whether the technique has been generally accepted in the relevant scientific community. 509 U.S. at 593-94, 113 S.Ct. at 2796-97. These factors “are not exhaustive and are intended to be.applied in a ‘flexible’ manner.” United Fire and Cas. Co., 704 F.3d at 1341 (quoting Kumho Tire Co., Ltd. v. Carmichael, 526 U.S. 137, 141, 119 S.Ct. 1167, 1171, 143 L.Ed.2d 238 (1999)). The focus of the inquiry contemplated by Rule 702 and Daubert “must be solely on principles and methodology, not on the conclusions that they generate.” Daubert, 509 U.S. at 595, 113 S.Ct. at 2797.
C. Discussion
A threshold question here is whether testimony of Williams’s cognitive impairment is even relevant to an alleged entrapment defense, as opposed to an insanity defense or a claim that Williams was not competent to stand trial. While the parties concede that it is relevant, and some non-binding case law seems to support that concession, this Circuit has never expressly held that it is. We do not answer that question here because there was no error in any event.
In this case, Dr. Fischer did not opine on Williams’s potential susceptibility to inducement—his given report and testimony focused only on the effect of Williams’s brain disease as it related to his competen
cy to stand trial. Similarly, Dr. Quinn expressed no pretrial opinion regarding Williams’s susceptibility to inducement to commit this crime, having testified only about his observations of Williams in connection with Williams’s competence to assist in his defense at trial. Therefore, the district court did not err in determining that, as a threshold matter, the testimony of Drs. Quinn and Fischer would only be admissible if they related to Dr. Rappa-port’s testimony concerning the relevant issue before the district court—Williams’s susceptibility to inducement.
However, although Dr. Rappaport testified at the Daubert hearing about Williams’s general loss of executive function and decision-making ability, his testimony did not tie the decline in Williams’s cognitive function to an accepted scientific methodology showing that that decline had caused Williams to be particularly susceptible to inducement at the time of his offense. Rather, Dr. Rappaport described the various tests that he had performed to determine Williams’s competency to stand trial, concluding that “the aberrations [he had found] in [Williams’s] mental status examination were ... most probably, ... organically based.” In other words, “[Williams] has some form of brain disease.”
Although Dr. Rappaport testified repeatedly that Williams was experiencing problems with executive functioning and decision-making due to that brain disease, Dr. Rappaport also testified that none of the tests that he had administered had been capable of assessing Williams’s mental condition on the night of his arrest in October of 2012, the relevant inquiry for. Williams’s entrapment defense. Dr. Rappa-port also testified that neither he nor anyone else would be able to say what the structure of Williams’s brain had been at that time. Further, Dr. Rappaport testified that there is no peer-reviewed body of literature that shows that, when a PET scan or MRI shows that the frontal lobe of a person’s brain is damaged—as the studies of Williams’s brain showed here—that person will be particularly susceptible to inducement to commit a crime, much less a sex crime with a 14-year-old victim.
Therefore, although Williams established that Dr. Rappaport had administered tests and evaluations that constituted accepted techniques for determining Williams’s mental status and mental problems at the time of his March 2013 examination, he did not establish how those tests and techniques determined (through a scientifically acceptable methodology) whether Williams had been particularly susceptible to inducement to commit this crime on or around October 2012. “[I]f an expert opinion does not have a ‘valid scientific connection to the pertinent inquiry* it should be excluded.” Boca Raton Cmty. Hosp., Inc. v. Tenet Health Care Corp., 582 F.3d 1227, 1232 (11th Cir. 2009) (quoting Daubert, 509 U.S. at 592, 113 S.Ct. at 2796).
Williams argues that Dr. Rappaport was prepared to testify about his own personal experience in treating patients with a diagnosed decline in executive function. However, neither Dr. Rappaport’s report nor his proffered testimony explained how his personal experience was relevant to the question of whether a decline in executive function caused one to be particularly susceptible to inducement to commit this crime.
Given that Dr. Rappaport did not identify a scientifically accepted method connecting his observations of Williams with a finding of susceptibility to inducement to commit this crime, the district court did not abuse its discretion in barring his testimony. And because Dr. Rappaport’s testimony was inadmissible, the district court
also did not abuse its discretion in barring the testimony of Drs. Fischer and Quinn, whose testimony was relevant only as a foundation for the testimony of Dr. Rappa-port.
IV. FAILURE TO GIVE A REQUESTED JURY INSTRUCTION
A. Applicable Law
This Court reviews “a district court’s refusal to give a requested jury instruction for abuse of discretion.” United States v. Carrasco. 381 F.3d 1237, 1242 (11th Cir. 2004). A district court’s failure to give a requested jury instruction is an abuse of discretion if the requested instruction “(1) was correct, (2) was not substantially covered by the charge actually given, and (3) dealt with some point in the trial so important that failure to give the requested instruction seriously impaired the defendant’s ability to conduct his defense.” United States v. Eckhardt, 466 F.3d 938, 947-48 (11th Cir. 2006). This Court “will not reverse a conviction unless [it] find[s] that issues of law were presented inaccurately or the charge improperly guided the jury in such a substantial way as to violate due process.” United States v. Perez-Tosta, 36 F.3d 1552, 1564 (11th Cir. 1994).
B. The Reasonable Doubt Instruction
Williams challenges the district court’s use of the pattern jury instruction on reasonable doubt, which again states, in relevant part: “The Government’s burden of proof is heavy, but it doesn’t have to prove a Defendant’s guilt beyond all possible doubt. The Government’s proof only has to exclude any reasonable doubt concerning the Defendant’s guilt.” We first put that sentence in the context of the full jury instruction on reasonable doubt, which states:
The indictment or formal charge against a Defendant isn’t evidence of guilt. The law presumes every Defendant is innocent. The Defendant does not have to prove his innocence or produce any evidence at all. The Government must prove guilt beyond a reasonable doubt. If it fails to do so, you must find the Defendant not guilty.
The Government’s burden of proof is heavy, but it doesn’t have to prove a Defendant’s guilt beyond all possible doubt. The Government’s proof only has to exclude any reasonable doubt concerning the Defendant’s guilt.
A “reasonable doubt” is a real doubt based on your reason and common sense after you’ve carefully and impartially considered all the evidence in the case.
“Proof beyond a reasonable doubt” is proof so convincing that you would be willing to rely and act on it without hesitation in the most important of your own affairs.
If you are convinced that the Defendant has been proved guilty beyond a reasonable doubt, say so. If you are not convinced, say so.
Williams argues that the use of the word “only” necessarily “lessens the Government’s burden of proof’ and “improperly implies that [Williams] has at least some burden to put forth a case against the Government’s.”
Williams’s argument is without merit. The charge first dearly shows that the government “must prove guilt beyond a reasonable doubt” and that, if it does not, the jury “must find the Defendant not guilty.” In the next paragraph, inclusion of the word “only,” particularly in the context of the entire jury instruction, does not alter the jury’s charge that the government bears a “heavy” burden to exclude “any” reasonable doubt as to the defendant’s guilt. Rather, it “guards against a
reading of ‘reasonable doubt’ that would require the Government to disprove all possible doubt—a burden higher than the law requires.” United States v. Rosin, 263 Fed.Appx. 16, 31 (11th Cir. 2008) (unpublished). The district court reiterated this reasoning in its discussion of Williams’s proposed jury instruction : “[T]he [pattern jury] instruction in its totality makes it clear that the jury doesn’t have to exclude all possible doubt, including hypothetical or fanciful doubt, but only a reasonable doubt, which is then [later] defined.”
Williams fails to show that the inclusion of this single word “only” would offer a materially different jury instruction than the one he proposes, much less that the pattern jury instruction would so mislead the jury as to deny Williams his constitutional right to due process. See Eckhardt, 466 F.3d at 948; Perez-Tosta, 36 F.3d at 1564. Accordingly, the district court did not abuse its discretion in refusing to give this proposed jury instruction.
C. The Entrapment Instruction
Williams also argues that the district court erred in failing to include several proposed revisions to the pattern entrapment jury instruction. These revisions highlighted Williams’s state of mind and diminished mental health. Williams argues that these revisions were “essential” to Williams’s entrapment defense because they highlighted physiological factors that made Williams unusually susceptible to inducement to commit this crime .and to government entrapment.
Although the district court originally agreed to include Williams’s proposed revisions, it ultimately did not because—following the district court’s exclusion of the proposed expert testimony about Williams’s psychological health—the revisions no longer applied. Indeed, the testimony of Drs. Rappaport, Fischer, and Quinn constituted the only possible expert testimony concerning the claim that Williams’s psychological health made him particularly susceptible to inducement to commit this crime. And it is undisputed that a lay witness could not testify to this specialized knowledge. See Fed. R. Evid. 702. Because the district court barred admission of these experts’ testimony following the, Daubert hearing, Williams thus presents no evidentiary support for the principle he seeks to highlight in his proposed revisions to the pattern entrapment jury instruction. Accordingly, the district court also did not err in refusing to include these pi’oposed revisions.
V. CONCLUSION
For all of these reasons, we affirm Williams’s conviction and sentence.
AFFIRMED.