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 3 or more matches.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the probability of getting 3 or more matches in the California Daily 4 lottery. This means we need to calculate the combined probability for x = 3 and x = 4.
2. Step 2: Refer to the provided table. The table lists the probabilities for different numbers of matching digits (x). Specifically, P(x=3) = 0.004 and P(x=4) = 0+.
3. Step 3: Add the probabilities for x = 3 and x = 4. Since the probability of x = 4 is given as 0+, it is effectively treated as 0 for practical purposes. Therefore, the total probability is P(x=3) + P(x=4).
4. Step 4: Write the formula for the calculation: \( P(x \geq 3) = P(x=3) + P(x=4) \). Substitute the values from the table into the formula.
5. Step 5: Interpret the result. The calculated probability represents the likelihood of getting 3 or more matches in the lottery. Ensure the addition is performed correctly to arrive at the final probability.

Key Concepts

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 essential for calculating the likelihood of specific events occurring.

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 3 or more matches, one must calculate the cumulative probability for 3 and 4 matches. This involves summing the probabilities of these outcomes, which allows for a comprehensive understanding of the likelihood of achieving a certain level of success in the lottery.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. In this context, to find the probability of getting 3 or more matches, one could alternatively calculate the probability of getting fewer than 3 matches (0, 1, or 2 matches) and subtract that from 1. This approach can simplify calculations and provide a clearer perspective on the desired outcome.