Arizona’s Pick 5 In one of Arizona’s lotteries, balls are numbered 1–35. 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 Arizona’s Pick 5 with one ticket.

Verified step by step guidance

1. Identify the total number of possible outcomes. Since 5 balls are selected from 35 without replacement and order does not matter, the total number of combinations is given by the combination formula: \(\displaystyle \binom{35}{5} = \frac{35!}{5! \times (35-5)!}\).
2. Understand that to win, your chosen 5 numbers must exactly match the 5 balls drawn. There is only 1 favorable outcome — the exact combination you picked.
3. Calculate the probability of winning by dividing the number of favorable outcomes by the total number of possible outcomes. This can be expressed as: \(\displaystyle P(\text{win}) = \frac{1}{\binom{35}{5}}\).
4. Interpret the result as the chance of your ticket matching the winning combination, which is a very small probability due to the large number of possible combinations.
5. If needed, you can leave the answer as a fraction or calculate the numerical value by evaluating the combination \(\binom{35}{5}\) using factorials or a calculator.

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 35 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 35 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.