Question

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

Mega Millions As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 75 and, in a separate drawing, you must also select the correct single number between 1 and 15. Find the probability of winning the jackpot. How does the result compare to the probability of being struck by lightning in a year, which the National Weather Service estimates to be 1/960,000?

Short answer

The probability of winning the jackpot is$\frac{1}{258,890,850}$.

The probability of winning the jackpot is much less than the probability of being struck by lightning in a year.

Step-by-step solution

Given information

Winning a lottery has two components. The first is to select the correct five numbers between 1 and 75, and the second is to select the correct single number between 1 and 15.

Define combination

When a certain number of units, say r, are to be chosen from a set of n units without replacement, then the combination rule is used to find the total number of ways in which the selections can be made.

The formula is shown below:

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

Here, the order of the selections has no importance.

Compute the number of ways to make five selections

The lottery is won when the correct selection of five numbers is made from 75 digits and one digit from 15 digits.

Case 1:

There are 75 numbers between 1 and 75.

The number of ways in which five numbers can be selected from 1 to 75 (in any order) is shown below:

$$\begin{gathered} {}^{75}{C}_5=\frac{75!}{\left(75-5\right)!\times 5!} \\ =\frac{75!}{70!\times 5!} \\ =17259390 \end{gathered}$$

The number of ways in which the correct five numbers can be selected 1.

The probability of selecting the correct five numbers from 1 to 75 is computed below:

$P\left(\text{correct}\text{5}\text{numbers}\right)=\frac{1}{17259390}$

Compute the number of ways to make one selection

Case 2:

The total number that can be selected between 1 and 15 is 15.

The number of ways in which one number can be selected from 1 to 15 (in any order) is shown below:

$$\begin{gathered} {}^{15}{C}_1=\frac{15!}{\left(15-1\right)!\times 1!} \\ =15 \end{gathered}$$

The number of ways in which the correct single number can be selected is 1.

The probability of selecting the correct numbers from 1 to 15 is computed below:

$P\left(\text{correct}\text{single}\text{number}\right)=\frac{1}{15}$

Compute the probability of winning the lottery

Let A be the event of winning the jackpot.

The probability of winning the jackpot is the product of the probabilities in cases 1 and 2. The calculation is shown below:

$$\begin{gathered} P\left(A\right)=\frac{1}{17259390}\times \frac{1}{15} \\ =\frac{1}{258890850} \end{gathered}$$

Therefore, the probability of winning the jackpot is$\frac{1}{258890850}$.

Compare the probability with the value

The probability of being struck by lightning is $\frac{1}{960,000}$.

The probability of winning the lottery is lesser than the probability of being struck by lightning, that is,

$\frac{1}{258,890,850}<\frac{1}{960,000}$

Therefore, the probability of winning the jackpot is much less than the probability of being struck by lightning in a year.