In Pennsylvania’s Cash 5 lottery, balls are numbered 1 to 43. Five balls are selected randomly, without replacement. The order in which the balls are selected does not matter. To win, your numbers must match the five selected. Determine your probability of winning Pennsylvania’s Cash 5 with one ticket.

Verified step by step guidance

1. Identify the total number of possible combinations of 5 balls drawn from 43 without replacement, where order does not matter. This is a combination problem, so use the formula for combinations: \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n=43\) and \(k=5\).
2. Calculate the total number of possible combinations: \(C(43, 5) = \frac{43!}{5!(43-5)!} = \frac{43!}{5! \times 38!}\).
3. Recognize that your ticket represents exactly one specific combination of 5 numbers out of all possible combinations.
4. The probability of winning with one ticket is the ratio of favorable outcomes (1 winning combination) to total possible outcomes (all combinations), so express the probability as \(P = \frac{1}{C(43, 5)}\).
5. Interpret this probability as the chance of your chosen 5 numbers matching the 5 numbers drawn, given the random selection without replacement and order not mattering.

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where order does not matter. In this lottery problem, since the order of the selected balls is irrelevant, we use combinations to calculate the total number of possible 5-ball sets from 43 balls.

Probability of a Single Event

The probability of a single event is the ratio of favorable outcomes to total possible outcomes. Here, the favorable outcome is matching the exact 5 balls drawn, and the total outcomes are all possible 5-ball combinations from 43 balls.

Without Replacement Sampling

Sampling without replacement means once a ball is drawn, it cannot be selected again. This affects the total number of possible combinations and ensures each combination is unique, which is essential for accurately calculating the probability.