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 of at least five out of the six subjects reverting to their first learned method is P(X ≥ 5) = P(X = 5) + P(X = 6). Using the binomial probability formula, we can calculate P(X = 5) and P(X = 6) and sum them to find the overall probability.

Step-by-step solution

Understand the binomial probability formula

The binomial probability formula is used to calculate the probability of a specific outcome (success or failure) in a fixed number of trials, each with the same probability of success. The formula is: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where - P(X = k) is the probability of k successes out of n trials, - C(n, k) is the number of combinations of n things taken k at a time (also written as n choose k), - p is the probability of success in each trial, and - n is the total number of trials. In this case, n = 6 (number of subjects tested), and p = 0.8 (probability of reverting to the first method learned).

Calculate the probability of exactly 5 subjects reverting

To find the probability of exactly 5 out of the 6 subjects reverting, we will use the binomial probability formula: P(X = 5) = C(6, 5) * (0.8)^5 * (1-0.8)^(6-5)

Calculate the probability of all 6 subjects reverting

To find the probability of all 6 subjects reverting, we will use the binomial probability formula again: P(X = 6) = C(6, 6) * (0.8)^6 * (1-0.8)^(6-6)

Calculate the probability of at least 5 subjects reverting

To find the probability of at least 5 subjects reverting, we need to sum the probabilities of exactly 5 and all 6 subjects reverting: P(X ≥ 5) = P(X = 5) + P(X = 6) Substitute the formulas from steps 2 and 3 and calculate the result.

Key concepts

Probability Theory

Understanding the crux of probability theory allows us to tackle real-world problems involving uncertainty and chance. In essence, this branch of mathematics deals with calculating the likelihood of various outcomes. When flipping a coin, for example, probability guides us in predicting whether it will land on heads or tails. In the context of our exercise regarding the subject reverting to the first method learned under stress, we use probability theory to determine the chances of this behavior occurring among six subjects.

Within probability theory, the binomial distribution is a cornerstone theory for events with two possible outcomes—often classified as 'success' or 'failure'. When considering our exercise, reverting to the first method is deemed a 'success', and the likelihood ('p') of this success is given as 0.8. We're essentially measuring how probable it is for this success to occur in multiple independent trials—each trial being one subject's response to the task under stress. Probability theory gives us a structured approach to quantify this uncertainty.

Combinatorics

Combinatorics is a fascinating field within mathematics focusing on counting, arranging, and combining objects according to certain rules. In probability calculations, combinatorics aids in determining the number of ways a particular event can occur. This has direct implications on our textbook problem, as it requires us to calculate how many possible ways five or all six subjects can revert to the original method when stressed.

Using combinatorial functions such as the 'combination'—denoted as C(n, k) or sometimes as 'n choose k'—we quantify the number of ways to choose 'k' successes out of 'n' trials. In this instance, C(6, 5) represents how many combinations can be made if five out of six subjects revert, while C(6, 6) is the scenario where all subjects revert. It's important to realize these combinatorial numbers are essential, multiplying factors in the binomial probability formula, contributing to the overall calculation. By mastering combinatorics, students gain an essential tool for handling various probability tasks.

Stress Effects on Learning

While mathematics often deals with numbers and abstract concepts, it's equally important to acknowledge the psychological aspects of human behavior—specifically, how stress affects learning and performance. Psychological studies indicate that under stress, individuals are more likely to revert to their most practiced or initial learning methods.

Our exercise implies that given sufficient mental strain, subjects tend to fall back on habits formed first. Understanding this type of response is crucial not only in studies of learning and memory but also in structuring educational experiences and interventions. For students, being aware of stress's impact on learning can promote strategies that maximize successful outcomes, such as practicing under test-like conditions or engaging in relaxation techniques before exams. While not typically covered in probability exercises, incorporating studies on stress and learning can enrich students' understanding of why certain probabilities, like the 0.8 chance of reverting to initially learned methods in the given problem, manifest in human behavior.