Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Using Probabilities for Significant Events

b. Find the probability of getting 2 or more matches.

검증된 단계별 안내

Step 1: Understand the problem. We are tasked with finding the probability of getting 2 or more matches in the California Daily 4 lottery. This means we need to calculate the sum of probabilities for x = 2, x = 3, and x = 4.

Step 2: Refer to the provided table. The table lists the probabilities for different numbers of matching digits (x). Specifically, P(x=2) = 0.049, P(x=3) = 0.004, and P(x=4) = 0+.

Step 3: Add the probabilities for x = 2, x = 3, and x = 4. Use the formula: \( P(x \geq 2) = P(x=2) + P(x=3) + P(x=4) \). Substitute the values from the table into this formula.

Step 4: Perform the addition. Combine the probabilities: \( P(x \geq 2) = 0.049 + 0.004 + 0+ \). Note that the probability for x = 4 is effectively zero (0+).

Step 5: Interpret the result. The sum represents the probability of getting 2 or more matches in the lottery. This is the final probability value, which can be used to understand the likelihood of this event occurring.

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주요 개념

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

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In this case, the table provides the probabilities for matching digits in the California Daily 4 lottery, indicating the likelihood of getting 0, 1, 2, 3, or 4 matches. Understanding this distribution is essential for calculating the probability of specific outcomes.

Cumulative Probability

Cumulative probability refers to the probability of a random variable being less than or equal to a certain value. To find the probability of getting 2 or more matches, one must calculate the cumulative probabilities for 0 and 1 matches and subtract them from 1. This concept is crucial for determining the likelihood of achieving at least a specified number of successes in a probability scenario.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. In this context, to find the probability of getting 2 or more matches, we can use the complement rule by calculating the probabilities of getting 0 or 1 matches and subtracting their sum from 1. This simplifies the calculation and provides a clear method for solving the problem.