Question

Matching suits A standard deck of playing cards consists of 52 cards with 13 cards in each of four suits: spades, diamonds, clubs, and hearts. Suppose you shuffle the deck thoroughly and deal 5 cards face-up onto a table.

a. What is the probability of dealing five spades in a row?

b. Find the probability that all 5 cards on the table have the same suit.

Short answer

The required answers are:

Part a) The probability is $0.0004952.$

Part b) The probability is$0.001981.$

Step-by-step solution

Part a) Step 1: Given information

Given that,

Number of cards in a deck $=52$

Number of cards in each suit$=13$

Part a) Step 2: Calculation

The probability of taking five spades can be calculated using the following formula:

$$\begin{gathered} P(fivespades)=\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}\times \frac{10}{49}\times \frac{9}{48} \\ =\frac{154440}{311875200} \\ =0.0004952 \end{gathered}$$

Therefore, the required probability is$0.0004952$

Part b) Step 1: Explanation

The likelihood of all five cards being of the same suit can be calculated as follows:

$$\begin{gathered} P(sameunits)=p(5spades)+P(5hearts)+P(5diamonds)+P(5clubs) \\ =\frac{33}{66640}+\frac{33}{66640}+\frac{33}{66640}+\frac{33}{66640} \\ =0.001981 \end{gathered}$$

Therefore, the required probability is$0.001981.$