In Exercises 5–36, express all probabilities as fractions.

Powerball As of this writing, the Powerball lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 69 and, in a separate drawing, you must also select the correct single number between 1 and 26. Find the probability of winning the jackpot.

Winning a lottery has two components.

The first is to select the correct five numbers from 1 and 69.

The second is to select the correct single number between 1 and 26.

When a certain number of units, say r, are chosen from aset of n units without replacement,the combination rule is used to find the total number of ways the selections can be made. Here, theorder of the selections has no significance.

The formula is shown below:

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

Let A be the event of winning the jackpot.

Case 1:

The total number of numbers between 1 and 69 is 69.

The number of ways in which five numbers can be selected from 1 to 69 (in any order) is shown below:

$$\begin{gathered} {}^{69}{C}_5=\frac{69!}{\left(69-5\right)!5!} \\ =11238513 \end{gathered}$$

The number of ways in which the correct five numbers can be selected is one.

The probability of selecting the correct five numbers from 1 to 69 is computed below:

$P\left(\text{correct}\text{5}\text{numbers}\right)=\frac{1}{11238513}$

Case 2:

The total number of numbers between 1 and 26 is 26.

The number of ways in which one number can be selected from 1 to 26 (in any order) is shown below:

$$\begin{gathered} {}^{26}{C}_1=\frac{26!}{\left(26-1\right)!1!} \\ =26 \end{gathered}$$

The number of ways in which the correct single number can be selected is one.

The probability of selecting the correct numbers from 1 to 26 is computed below:

$P\left(\text{correct}\text{single}\text{number}\right)=\frac{1}{26}$

The probability of winning the jackpot is the product of the probabilities in cases 1 and 2.

The calculation is shown below:

$$\begin{gathered} P\left(A\right)=\frac{1}{11238513}\times \frac{1}{26} \\ =\frac{1}{292201338} \end{gathered}$$

Therefore, the probability of winning the jackpot is equal to$\frac{1}{292201338}$.