Question

A subject is taught to do a task in two different ways. Studies have shown that when subjected to mental strain and asked to perform the task, the subject most often reverts to the method first learned, regardless of whether it was easier or more difficult. If the probability that a subject returns to the first method learned is .8 and six subjects are tested, what is the probability that at least five of the subjects revert to their first learned method when asked to perform their task under stress?

Short answer

Answer: The probability is approximately 0.65536.

Step-by-step solution

Understand the binomial probability formula

The binomial probability formula is used to find the probability of a specific number of successes in a fixed number of trials of a discrete probability experiment, where each trial has only two possible outcomes (success or failure). The formula is defined as: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where n = number of trials k = number of successes p = probability of success C(n, k) = combination of n things taken k at a time (n! / (k!(n-k)!)) In our case, a "success" means a subject reverts to their first learned method, with a probability of success p = 0.8, and we have n = 6 trials (subjects) in our experiment.

Calculate the probability for exactly five subjects

First, we will find the probability of exactly five subjects reverting to their first learned method. Here, we have k = 5 successes. Using the binomial probability formula: P(X = 5) = C(6, 5) * 0.8^5 * (1 - 0.8)^(6-5) P(X = 5) = (6! / (5!(6-5)!)) * 0.8^5 * 0.2^1 P(X = 5) = (720 / (120*1)) * 0.8^5 * 0.2 P(X = 5) = 6 * 0.32768 * 0.2 P(X = 5) ≈ 0.39322 The probability that exactly five subjects revert to their first learned method is approximately 0.39322.

Calculate the probability for all six subjects

Now, we find the probability that all six subjects revert to their first learned method. In this case, k = 6 successes. Using the binomial probability formula: P(X = 6) = C(6, 6) * 0.8^6 * (1 - 0.8)^(6-6) P(X = 6) = (6! / (6!(6-6)!)) * 0.8^6 * 1 P(X = 6) = 1 * 0.262144 * 1 P(X = 6) ≈ 0.26214 The probability that all six subjects revert to their first learned method is approximately 0.26214.

Calculate the probability that at least five subjects revert to their first learned method

To find the probability that at least five subjects revert to their first method, we simply add the probabilities of the two cases (exactly five subjects and all six subjects) that we calculated in Steps 2 and 3: P(at least 5 subjects) = P(X = 5) + P(X = 6) P(at least 5 subjects) ≈ 0.39322 + 0.26214 P(at least 5 subjects) ≈ 0.65536 The probability that at least five of the six subjects revert to their first learned method when asked to perform their task under stress is approximately 0.65536.

Key concepts

Probability of Success

In any probability experiment, understanding the concept of "probability of success" is crucial. This term refers to the likelihood of a specific outcome happening in a trial. In our exercise, the probability of success is 0.8, meaning there's an 80% chance that a subject will revert to their first learned method.

Probability of success is often denoted by the letter 'p' in formulas. This probability remains constant across all trials in a binomial distribution, which is important in ensuring consistency and reliability of results. To illustrate:

• If p = 0.8, then each subject has an 80% chance of reverting to the first method.
• Therefore, as the number of subjects increases, the probability of achieving more successes also gets higher.

Assessing the probability of success allows us to determine the expected rate of occurrence of the desired outcome, helping in the practical prediction of results.

Discrete Probability Experiment

A discrete probability experiment involves trials with specific, distinguishable outcomes. In the case of our exercise, we are dealing with binomial trials, each having two possible results: a subject either reverts to their first learned method or doesn't.

Discrete probability experiments embrace the notion of independence. This means the outcome of one trial does not affect another. Here, the actions of one subject do not influence the others. Important features include:

• Fixed number of trials (n): In our problem, there are six subjects tested, which comprise six distinct trials.
• Two outcomes per trial: Success (reverting to the first method) or failure (not reverting). Both with known probabilities.
• Consistency: Each trial's success probability, 0.8, remains unchanged.

By analyzing these experiments using the binomial framework, we can generate predictions on the outcomes' probabilities.

Combinatorial Formula

The combinatorial formula plays a vital role in calculating probabilities in binomial distributions. It helps to determine the number of ways to select "k" successes from "n" trials. This is denoted by the combination symbol, C(n, k).

The formula is written as: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where "n!" (n factorial) represents the product of all positive integers up to "n".

In our exercise:

• When calculating for exactly five successes, we use C(6, 5), which equals 6.
• For all six successes, it's C(6, 6), resulting in 1 way.

This formula is crucial as it allows us to weigh each potential outcome by the number of ways it can occur. Understanding this process is essential in mastering probability calculations and predicting event probabilities accurately.