Question

Suppose five cards are drawn from a deck.

What is the probability of obtaining a flush (five cards of the same suit)?

Answer 1

Explanation:

Step-$1$: Total number of outcomes:

Five cards are drawn from the deck of $52$ cards.

Then total no of outcomes is

$$\begin{gathered} {}^{52}C_5=\frac{52!}{5!·\left(52-5\right)!} \\ =\frac{52!}{5!·47!} \\ =\frac{52\times 51\times 50\times 49\times 48}{5\times 4\times 3\times 2} \\ =2598960 \end{gathered}$$

Step-2: Favorable outcomes:

The five cards drawn are of same suit from four suits

Therefore possible outcomes are

$$\begin{gathered} {}^{4}C_1{}^{13}C_5=\frac{4!}{3!}\times \frac{13!}{5!·\left(13-5\right)!} \\ =4\times \frac{13!}{5!·8!} \\ =\frac{4\times 13\times 12\times 11\times 10\times 9\times 8!}{5!\times 8!} \\ =\frac{4\times 13\times 12\times 11\times 10\times 9}{5\times 4\times 3\times 2} \\ =4\times 13\times 11\times 9 \\ =5148 \end{gathered}$$

Step-$3$: Probability of flush card:

the probability of obtaining a flush is

$$\begin{gathered} =\frac{no.ofpossibleoutcomes}{totalnoofoutcomes} \\ =\frac{5148}{2598960} \\ \cong 0.00198 \end{gathered}$$

Therefore, the probability of obtaining a flush is $0.00198$.