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Aldrich Ames is a convicted traitor who leaked American secrets to a foreign power. Yet Ames took routine lie detector tests and each time passed them. How can this be done? Recognizing control questions, employing unusual breathing patterns, biting one's tongue at the right time, pressing one's toes hard to the floor, and counting backwards by 7 are countermeasures that are difficult to detect but can change the results of a polygraph examination (Source: Lies! Lies!! Lies!!! The Psychology of Deceit, by C. V. Ford, professor of psychiatry, University of Alabama). In fact, it is reported in Professor Ford's book that after only 20 minutes of instruction by "Buzz" Fay (a prison inmate), \(85 \%\) of those trained were able to pass the polygraph examination even when guilty of a crime. Suppose that a random sample of nine students (in a psychology laboratory) are told a "secret" and then given instructions on how to pass the polygraph examination without revealing their knowledge of the secret. What is the probability that (a) all the students are able to pass the polygraph examination? (b) more than half the students are able to pass the polygraph examination? (c) no more than four of the students are able to pass the polygraph examination? (d) all the students fail the polygraph examination?

Step by step solution

01

Understand the Scenario

The scenario involves determining the probability of students passing a polygraph examination after receiving training. The success rate of passing the test after training is 85% for each student. This can be modeled using a binomial distribution because each student's result is independent and has two outcomes: pass or fail.

02

Define the Binomial Distribution

The problem involves a binomial distribution with parameters: \( n = 9 \) (number of trials or students) and \( p = 0.85 \) (probability of success for each trial). Thus, we have: \[ X \sim B(n=9, p=0.85) \] where \( X \) is the random variable representing the number of students passing the test.

03

Determine the Probability for Part (a)

For part (a), we need the probability that all students pass, which equates to 9 students passing. Use the binomial probability formula:\[ P(X = 9) = \binom{9}{9} (0.85)^9 (0.15)^0 \]Calculate this to find the probability.

04

Determine the Probability for Part (b)

Part (b) asks for the probability that more than half of the students pass. More than half means at least 5 students passing (5 to 9 inclusive). Calculate:\[ P(X \geq 5) = \sum_{x=5}^{9} P(X = x) \]Use the binomial probability formula for each value from 5 to 9, and sum these probabilities.

05

Determine the Probability for Part (c)

Part (c) involves calculating the probability that no more than 4 students pass. This is given by:\[ P(X \leq 4) = \sum_{x=0}^{4} P(X = x) \]Again, use the binomial probability formula for each value from 0 to 4 and sum the probabilities.

06

Determine the Probability for Part (d)

For part (d), calculate the probability that all students fail. If all students fail, then 0 students pass:\[ P(X = 0) = \binom{9}{0} (0.85)^0 (0.15)^9 \]Calculate this probability to find the result.

07

Calculate Each Probability

Use the binomial probability formula and calculate each required probability.- For part (a): Compute \( P(X = 9) \).- For part (b): Sum while calculating \( P(X = 5), P(X = 6), P(X = 7), P(X = 8), P(X = 9) \).- For part (c): Sum while calculating \( P(X = 0), P(X = 1), P(X = 2), P(X = 3), P(X = 4) \).- For part (d): Compute \( P(X = 0) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability

When we talk about probability, we are discussing the likelihood of an event occurring. Probability values range from 0 to 1, where 0 indicates the event will definitely not happen, and 1 indicates that it definitely will. In the context of a polygraph examination -- where students are either passing or failing after training -- the probability of passing is either 85% or 0.85 for each student.
This scenario is a classic example of binomial distribution where each student represents an independent trial. The probability can be calculated for different outcomes such as all students passing, more than half of them passing, etc. Each possibility is a separate event, and by employing probability formulas, we can determine the likelihood of such events. Understanding these probabilities allows experimenters to predict outcomes more accurately based on given data.

Polygraph Examination

A polygraph examination, commonly known as a lie detector test, is utilized to determine if an individual is being deceptive. It measures physiological responses such as breathing patterns, heart rate, and sweat gland activity. These responses are believed to vary when a person is lying. However, the effectiveness of polygraph tests has been a subject of debate in psychology.
In this exercise, students learn techniques to alter their physiological responses to pass these examinations even when guilty of deception. These countermeasures make interpreting the results challenging, reducing the reliability of polygraph tests. It's important to note that while polygraphs can indicate physiological changes, they cannot definitively prove lies or truths.

Psychology Experiment

Conducting psychology experiments involves creating scenarios to test hypotheses about human behavior and mental processes. The setup with the students trying to pass a polygraph test is a form of such an experiment. Here, the hypothesis is about the effectiveness of training on mastering the polygraph exam regardless of truthfulness.
By instructing the students and running such experiments, psychologists aim to understand the role of mental preparation and strategy on physical responses. Issues like the ethics and implications of teaching such countermeasures can also be explored, allowing for a comprehensive view of how psychological principles can affect real-world situations.

Statistical Analysis

Statistical analysis forms the backbone of deriving meaningful conclusions from experimental data. In this context, it involves using the binomial distribution to calculate probabilities of different outcomes -- such as how many students will pass the polygraph test after training.
Each student's attempt at the polygraph test can be seen as a trial, and statistical analysis allows us to sum up the probabilities of various outcomes, like all students passing or more than half passing. Through this method, we can statistically infer how effective the training is, despite its potential ethical issues, based on observed data versus expected probabilities.