Question

Four cards are drawn from a deck of $52$ cards, the probability of only two cards being spade is......

• A. $\frac{27}{128}$
• B. $\frac{54}{128}$
• C. $\frac{81}{128}$
• D. $\frac{31}{128}$

Answer 1

The correct option is B $\frac{54}{128}$

Let $X$: be the number of spade cards.

Drawing is a Bernoulli trial.

So, $X$ has the binomial distribution, $P(X=x){=}^n{C}_x{q}^{n-x}{p}^x$

where n = number of cards drawn = 4, p = probability of getting a spade card = $\frac{13}{52}=\frac{1}{4}$

Hence, $q=1-p=1-\frac{1}{4}=\frac{3}{4}$

Hence $P(X=x){=}^4{C}_x{\left(\frac{3}{4}\right)}^{4-x}{\left(\frac{1}{4}\right)}^x$

Hence the probability of only two cards being spade is $P(X=2){=}^4{C}_2{\left(\frac{3}{4}\right)}^{4-2}{\left(\frac{1}{4}\right)}^2=6\times {\left(\frac{3}{4}\right)}^2{\left(\frac{1}{4}\right)}^2=6\times \frac{9}{16}\times \frac{1}{16}=\frac{27}{128}$