Question

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

Answer: The probability of getting three matches in three consecutive coin tosses is 1/8 (12.5%). The probability of all six tosses being tails is 1/64 (1.56%). Although the probability of getting three matches is low, it does not necessarily suggest collusion between two students. Other factors, such as similarity in wrong answers and the relationship between the students, should be considered before drawing any conclusions.

Step-by-step solution

Calculate the probability of a match in a single toss

In a single toss, there are 4 possible outcomes: HH, HT, TH, and TT. Two of these outcomes are matches (HH and TT). So, the probability of a match in a single toss is: P(match) = Number of matches outcomes / Total number of outcomes P(match) = 2 / 4 = 1 / 2

Calculate the probability of three matches

Since the coin tosses are independent events, we can find the probability of three matches by multiplying the probabilities of matches in each toss. So the probability of three matches is: P(3 matches) = P(match) * P(match) * P(match) = (1/2) * (1/2) * (1/2) = 1 / 8

Calculate the probability of getting tails in a single toss

In a single coin toss, there are two possible outcomes: heads (H) and tails (T). The probability of getting tails is: P(Tails) = Number of tails outcomes / Total number of outcomes P(Tails) = 1 / 2

Calculate the probability of all six tosses being tails

Again, since the tosses are independent, we can find the probability of all six tosses being tails by multiplying the probabilities in each toss. P(All 6 tails) = P(Tails) * P(Tails) * P(Tails) * P(Tails) * P(Tails) * P(Tails) = (1/2)^6 = 1 / 64

Analyze if low probability suggests collusion

From part a, we found that the probability of getting three matches is 1/8, which means there is a 12.5% chance of two independent coin tosses having three matches. The low probability might seem suspicious at first, but it is important to consider other factors before concluding collusion between the two students. It is not rare for independent events to produce matching outcomes occasionally, and a 12.5% probability is not exceptionally low. So, it would not be correct to suggest collusion solely based on the probability found in part a. Other factors, like the similarity in the wrong answers and the relationship between the students, should be taken into account before making any conclusions.

Key concepts

Coin Toss

A coin toss is a simple yet fascinating probability exercise. When you toss a coin, there are two possible outcomes: heads (H) or tails (T). For each toss, the chance of the coin landing on heads or tails is equal, which makes it a perfect example of a 50-50 chance. Coin tossing is considered a model for random events, meaning each toss is independent of the previous one. The outcome of one toss does not affect the result of the next. Understanding the nature of a coin toss is crucial when analyzing multiple tosses in sequence, like the exercise where two men toss coins to achieve matching results.

Independent Events

Independent events are those where the outcome of one event does not influence the outcome of another. Each coin toss in the exercise is independent, as the result of one coin has no bearing on the next. This concept is essential in calculating probabilities because it allows you to multiply the probabilities of individual events to find the total probability of a sequence of independent events occurring.For example, when calculating the probability of three consecutive matches in the coin toss exercise, each toss remains independent, so you multiply the probability of a single match \( \left( \frac{1}{2} \right) \) three times: \( \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{8} \). This demonstrates how understanding independence helps in solving complex probability problems.

Match Probability

Match probability refers to the chance that two independent events will produce the same outcome, like both coins landing heads or tails in the same toss. In the given problem, a match occurs when both men have the same result, either both heads (HH) or both tails (TT). Since there are four possible outcomes per toss (HH, HT, TH, TT), and two are considered matches, the probability of a match per toss is \( \frac{2}{4} = \frac{1}{2} \).Understanding match probability is essential when calculating more complex sequences of events. For instance, to find the probability of achieving three matches in a row, you simply multiply the probability of each match, considering the independence of tosses. This leads to a probability of \( \frac{1}{8} \), highlighting how probabilities compound over multiple independent events.

Collusion Analysis

Collusion analysis involves looking at whether a pattern of results, like the matching answers in exams, suggests cooperation between participants. In the example exercise, we see a slight concern about collusion because two students provide matching answers more often than expected. However, while the probability of getting three matches in a row is relatively low at 12.5%, it is not so low as to be highly suspicious by itself. To assess the possibility of collusion, other factors should be considered. These might include:

• Similarity in incorrect answers, not just matches.
• Any known relationship or previous cooperation between the individuals.
• Additional statistical patterns over multiple tests or conditions.

A simple probability calculation can raise questions but is just one piece of the puzzle when investigating potential collusion.