Unusual Events In Exercises 37 and 38, find the indicated probabilities. Then determine if the event is unusual. Explain your reasoning.

Rock-Paper-Scissors The probability of winning a game of rock-paper-scissors is 1/3. You play nine games of rock-paper-scissors. Find the probability that the number of games you win is (c) less than two.

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Step 1: Recognize that this is a binomial probability problem because there are a fixed number of trials (9 games), two possible outcomes for each trial (win or not win), and the probability of success (winning) is constant at 1/3.

Step 2: Define the random variable X as the number of games won. X follows a binomial distribution with parameters n = 9 (number of trials) and p = 1/3 (probability of success). The probability mass function for a binomial distribution is given by: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'k' is the number of successes.

Step 3: To find the probability that the number of games won is less than 2, calculate P(X < 2). This is equivalent to P(X = 0) + P(X = 1). Use the binomial probability formula for each term: P(X = 0) = (9 choose 0) * (1/3)^0 * (2/3)^9 and P(X = 1) = (9 choose 1) * (1/3)^1 * (2/3)^8.

Step 4: Compute the binomial coefficients (n choose k) for each term. For P(X = 0), (9 choose 0) = 1. For P(X = 1), (9 choose 1) = 9. Substitute these values into the respective probability expressions.

Step 5: Add the probabilities P(X = 0) and P(X = 1) to get P(X < 2). Finally, compare the result to a threshold (e.g., 0.05) to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of games like rock-paper-scissors, the probability of winning a single game is 1/3, meaning that in a large number of games, you would expect to win about one-third of the time. Understanding probability is essential for calculating the chances of winning multiple games.

Binomial Distribution

The binomial distribution is a statistical distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, playing nine games of rock-paper-scissors can be modeled as a binomial distribution where the number of trials is 9 and the probability of winning each trial is 1/3. This concept is crucial for calculating the probability of winning fewer than two games.

Unusual Events

An event is considered unusual if its probability is significantly low, often defined as less than 5%. In the context of the question, after calculating the probability of winning less than two games, you would compare this probability to the threshold of 5% to determine if the outcome is unusual. This concept helps in assessing the significance of the results in statistical analysis.