Kentucky Pick 4 In the Kentucky Pick 4 lottery game, you can pay \$1 for a “straight” bet in which you select four digits with repetition allowed. If you buy only one ticket and win, your prize is \$2500.


a. If you buy one ticket, what is the probability of winning?

Verified step by step guidance

1. Understand the problem: In the Kentucky Pick 4 lottery, you select four digits (0-9) with repetition allowed. The total number of possible outcomes is the total number of 4-digit combinations where each digit can range from 0 to 9.
2. Calculate the total number of possible outcomes: Since each digit has 10 possible values (0 through 9), and there are 4 digits, the total number of outcomes is given by \( 10^4 \).
3. Determine the number of favorable outcomes: To win, the exact 4-digit combination you selected must match the winning combination. Therefore, there is only 1 favorable outcome.
4. Calculate the probability of winning: Probability is defined as the ratio of favorable outcomes to total outcomes. Use the formula \( P(\text{winning}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \). Substitute the values from the previous steps.
5. Simplify the probability expression: Simplify the fraction \( \frac{1}{10^4} \) to express the probability in its simplest form.

Key Concepts

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 the Kentucky Pick 4 lottery, the probability of winning with a single ticket can be calculated by determining the total number of possible outcomes and the number of favorable outcomes (winning combinations).

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In the Kentucky Pick 4 game, since digits can be repeated, the total number of possible combinations can be calculated using the formula for permutations with repetition, which is essential for determining the total outcomes in the lottery.

Expected Value

Expected value is a key concept in statistics that represents the average outcome of a random event when considering all possible outcomes and their probabilities. In the lottery context, understanding the expected value can help players assess whether the potential prize justifies the cost of the ticket, given the low probability of winning.