Question

Pennsylvania Cash 5 In the Pennsylvania Cash 5 lottery, winning the top prize requires that you select the correct five different numbers between 1 and 43 (in any order). What is the probability of winning the top prize? Express the answer as a fraction.

Short answer

The probability of selecting the correct five numbers is $\frac{1}{962598}$.

Step-by-step solution

Given Information

Five number cards are selected from 1 to 43.

The number of the top prize card combination is 1.

Define the probability of an event and the combination rule

The formula for the probability of an event A is:

$P\left(A\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

Often the total number of outcomes and favorable outcomes are computed using the combination rule.

The formula of the combination rule for selecting ritems from n is:

${}^n{C}_r=\frac{n!}{\left(n-r\right)!\times r!}$

Compute the total outcomes

The number of ways to select five cards from 43 is:

$$\begin{gathered} {}^{43}{C}_5=\frac{43!}{\left(43-5\right)!5!} \\ =962598 \end{gathered}$$

Compute the probability of correct five numbers

Define event E as the probability of winning the top prize.

The probability of winning the top prize is:

$P\left(E\right)=\frac{1}{962598}$

Thus, the probability of winning the top prize is $\frac{1}{962598}$.