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

a. Find the probability of getting exactly 3 matches.

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Step 1: Understand the problem. The goal is to find the probability of getting exactly 3 matches in the California Daily 4 lottery. The table provided lists the probabilities for different numbers of matching digits (x).

Step 2: Locate the relevant probability in the table. The table shows the probability P(x) for each number of matching digits. For x = 3 (exactly 3 matches), the corresponding probability is 0.004.

Step 3: Interpret the probability. The value 0.004 represents the likelihood of selecting 3 digits that match the drawn digits in the same order during a 'straight' bet.

Step 4: Verify the context. Ensure that the table and problem description align with the calculation. The table explicitly provides the probability for x = 3, so no further computation is needed.

Step 5: Conclude the process. The probability of getting exactly 3 matches is directly obtained from the table as 0.004. This concludes the solution process.

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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 random variable x represents the number of matching digits in the lottery, and the table provides the probabilities for each possible outcome (0 to 4 matches). Understanding this distribution is crucial for calculating the likelihood of specific events, such as getting exactly 3 matches.

Random Variable

A random variable is a numerical outcome of a random phenomenon. In the context of the lottery question, the random variable x indicates the number of digits that match the drawn numbers in the same order. Recognizing how random variables function helps in analyzing the probabilities associated with different outcomes in probabilistic scenarios.

Calculating Probabilities

Calculating probabilities involves determining the likelihood of a specific event occurring based on a probability distribution. For this question, to find the probability of getting exactly 3 matches, one would refer to the provided table and identify the corresponding probability value, which is 0.004. This process is fundamental in statistics for making informed predictions about random events.