91Ó°ÊÓ

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

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.

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 $\frac{1}{2598960}$.

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 $\frac{1}{28}$.

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.

c.

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

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

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