Question

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

• A. $\frac{1}{256}$
• B. $\frac{1}{56}$
• C. $\frac{1}{64}$
• D. $\frac{31}{256}$

Answer 1

The correct option is A $\frac{1}{256}$

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 all being spade is $P(X=4){=}^4{C}_4{\left(\frac{3}{4}\right)}^{4-4}{\left(\frac{1}{4}\right)}^4={\left(\frac{1}{4}\right)}^4=\frac{1}{256}$