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Chapter 4: Problem 121

Two men each toss a coin. They obtain a "match" if either both coins are heads or both are tails. Suppose the tossing is repeated three times. a. What is the probability of three matches? b. What is the probability that all six tosses (three for each man) result in tails? c. Coin tossing provides a model for many practical experiments. Suppose that the coin tosses represent the answers given by two students for three specific true-false questions on an examination. If the two students gave three matches for answers, would the low probability found in part a suggest collusion?

Short Answer

Expert verified
Answer: The probability of three matches happening is 1/8 (12.5%), while the probability of all six tosses resulting in tails is 1/64 (1.56%). The low probability of three matches occurring (1/8) suggests that there is a 12.5% chance that two students randomly guessing on three true-false questions would have all three matches. While this is a low probability, it is not strong evidence for collusion in an academic context. Other factors, such as overall response patterns or prior knowledge of collaboration, should be considered before concluding collusion.

Step by step solution

01

Identify the sample space and the probability of each outcome for a single toss

For a single coin toss, there are two possible outcomes: heads (H) or tails (T). Since both men are tossing a coin, there will be four possible outcomes for a single round: 1. HH 2. HT 3. TH 4. TT Since the coins are fair and the tosses are independent, the probability of each outcome happening is 1/2 * 1/2 = 1/4.
02

Calculate the probability of a match happening

For a single round, there are two outcomes that can be considered as matches: HH and TT. Since each outcome has a probability of 1/4, the probability of a match happening in a single round is 1/4 + 1/4 = 1/2.
03

Calculate the probability of three matches happening

Since the coin tosses are independent, the probability of three matches happening in three rounds is the product of the probabilities of a match happening in each round. Therefore, the probability of three matches happening = (1/2) * (1/2) * (1/2) = 1/8.
04

Calculate the probability of all six tosses resulting in tails

The probability of a tail occurring in a single toss is 1/2. Since each toss is independent, the probability of all six tosses resulting in tails is: (1/2)^6 = 1/64.
05

Evaluate the probability of three matches in the context of true-false questions

The low probability (1/8) of three matches can be interpreted as an 87.5% chance that they will not have three matches. In the context of true-false questions, this means that if two students randomly guessed on three specific questions, they would have a 12.5% chance of getting all three matches. While this is a low probability, it is not necessarily suitable as strong evidence for collusion in an academic context. There may be other factors to consider, such as similarities in the overall pattern of responses or prior knowledge of a pattern of collaboration between the students. However, on its own, a probability of 1/8 for three matches does not constitute definitive proof of collusion.

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