Question

New Lottery Game In the Monopoly Millionaires’ Club lottery game, you pay $5 and select five different numbers between 1 and 52, and then a sixth number between 1 and 28 is randomly assigned to you. Winning requires that your five numbers match those drawn (in any order), and then your sixth assigned number must also match.

a. What is the probability of selecting the correct five numbers between 1 and 52?

b. What is the probability of getting the correct sixth number that is assigned to you?

c. What is the probability of selecting the correct five numbers and getting the correct sixth number?

Short answer

a. The probability of selecting the correct five numbers is equal to $\frac{1}{2598960}$.

b. The probability of getting the correct sixth number is equal to $\frac{1}{28}$.

c. The probability of selecting the correct five numbers and getting the correct sixth number is equal to $\frac{1}{72770880}$.

Step-by-step solution

Given information

In a lottery, five numbers are drawn from 1 to 52, and one number is allotted from 1 to 28.

Describe combination rule

The number of ways in which r items can be selected from n items without replacement when the order of the selected items does not matter is computed by the combination rule.

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

Further, the probability is computed as favorable counts of outcomes for an event to the total counts.

Compute the probability using the combination rule

a.

Let A be the event of selecting the correct five numbers.

The total numbers to choose from is 52

The numbers to be selected are 5.

The number of ways in which five numbers can be selected without replacement in any order is equal to:

$$\begin{gathered} {}^{52}{C}_5=\frac{52!}{\left(52-5\right)!5!} \\ =2598960 \end{gathered}$$

The number of ways in which the correct five numbers can be chosen = 1

The probability of selecting the correct five numbers is equal to:

$P\left(A\right)=\frac{1}{2598960}$

Therefore, the probability of selecting the correct five numbers is equal to $\frac{1}{2598960}$.

Compute probability for correct selection

b.

Let B be the event of getting the correct sixth number.

The total numbers to choose from = 28

The numbers to be selected = 1

The number of ways in which the sixth number can be selected without replacement in any order is equal to:

$$\begin{gathered} {}^{28}{C}_1=\frac{28!}{\left(28-1\right)!1!} \\ =28 \end{gathered}$$

The number of ways in which the correct sixth number can be obtained = 1

The probability of getting the correct sixth number is equal to:

$P\left(B\right)=\frac{1}{28}$

Therefore, the probability of getting the correct sixth number is equal to $\frac{1}{28}$.

Describe the multiplication rule

The multiplication rule is applied when more than one event occurs together. If the events are independent, the probability of joint occurrence is equal to the product of probabilities for each of the events.

Compute probability using the multiplication rule

c.

The probability of selecting the correct five numbers and getting the correct sixth number is equal to:

$$\begin{gathered} P\left(AandB\right)=P\left(A\right)\times P\left(B\right) \\ =\frac{1}{2598960}\times \frac{1}{28} \\ =\frac{1}{72770880} \end{gathered}$$

Therefore, the probability of selecting the five numbers and getting the correct sixth number is equal to $\frac{1}{72770880}$.