Question

Wild Card Lottery The Wild Card lottery is run in the states of Idaho, Montana, North Dakota, and South Dakota. You must select five different numbers between 1 and 33, then you must select one of the picture cards (Ace, King, Queen, Jack), and then you must select one of the four suits (club, heart, diamond, spade). Express all answers as fractions.

a. What is the probability of selecting the correct five numbers between 1 and 33?

b. What is the probability of selecting the correct picture card?

c. What is the probability of selecting the correct suit?

d. What is the probability of selecting the correct five numbers, the correct picture card, and the correct suit?

Short answer

a. The probability of selecting the correct five numbers is $\frac{1}{237,336}$ .

b. The probability of selecting the correct picture card is $\frac{1}{4}$ .

c. The probability of selecting the correct suit is $\frac{1}{4}$ .

d. The probability of selecting the correct five numbers, the correct picture card, and, the correct suit is $\frac{1}{3,797,376}$ .

Step-by-step solution

Given Information

Fiver numbers are to be selected from 1 to 33.

One each from picture cards and suit cards is to be selected from four.

Define the probability of an event

The probability of an event is defined as:

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

Favorable outcomes would support the occurrence of event A out of the total number of outcomes.

Define combination 

Combination is the measure to compute the number of ways to pick a certain number of items from a set of values.

The formula to select r items from n using the combination rule is:

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

Compute the probability of the correct five numbers

a.

The number of ways in which five numbers can be chosen from 1 to 33 is:

$$\begin{gathered} {}^{33}{C}_5=\frac{33!}{\left(33-5\right)!\times 5!} \\ =237336 \end{gathered}$$

Thus, the number of ways to pick five number cards from 1 to 33 is 237336.

The number of correct combinations is 1.

The probability of event E, i.e., selecting the correct five numbers from 1 to 33 is:

$P\left(E\right)=\frac{1}{237336}$

Thus, the probability of selecting the correct five numbers is $\frac{1}{237336}$ .

Compute the probability of correct picture card

b.

The number of cards that is correct is 1.

The total number of picture cards is 4.

The probability of event F, i.e., selecting the correct picture card is:

$P\left(F\right)=\frac{1}{4}$

Thus, the probability of selecting the correct picture card is $\frac{1}{4}$.

Compute the probability of the correct suit card

c.

The number of suit cards that are correct is 1.

The total number of suit cards is 4.

The probability of event G, i.e., selecting the correct suit card is:

$P\left(G\right)=\frac{1}{4}$

Thus, the probability of selecting the correct suit card is $\frac{1}{4}$.

 Step 7: Compute the probability of the occurrence of all events together

d.

When multiple events occur together, the probability of co-occurrence is computed as the product of all these events. This is stated as the multiplication rule.

The probability of selecting a five number card, picture card, and suit card is computed as follows:

$$\begin{gathered} P\left(E\text{and}F\text{and}G\right)=P\left(E\right)\times P\left(F\right)\times P\left(G\right) \\ =\frac{1}{237336}\times \frac{1}{4}\times \frac{1}{4} \\ =\frac{1}{3797376} \end{gathered}$$

Thus, the probability of selecting a five number card, picture card, and suit card is $\frac{1}{3797376}$.