Question

If 6 cards are drawn at random from a standard deck of 52 cards, what is the probability that exactly 2 of the cards are spades?

Answer 1

2/52. hope this helps

Answer 2

Answer:  The required probability is 31.51%.

Step-by-step explanation:  Given that 6 cards are drawn at random from a standard deck of 52 cards.

We are to find the probability that exactly two of the cards are spades.

Let S denote the sample space of drawing 6 cards from a deck of 52 cards and let A denote the event that exactly two of them are spades.

Then, we have

$n(S)=^{52}C_6=\dfrac{52!}{6!(52-6)!}=\dfrac{52\times51\times50\times49\times48\times46!}{6\times5\times4\times3\times2\times1\times46!}=20358520,\\\\\\n(A)\\\\\\=^{13}C_2\times^{39}C_4~~~~~~~~~~~~~~~~[\textup{there are 13 spades in the deck}]\\\\\\=\dfrac{13!}{2!(13-2)!}\times\dfrac{39!}{4!(39-4)!}\\\\\\=\dfrac{13\times12\times11!}{2\times1\times11!}\times\dfrac{39\times38\times37\times36\times35!}{4\times3\times2\times1\times35!}\\\\\\=78\times82251\\\\=6415578.$

Therefore, the probability of event A is given by

$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{6415578}{20358520}=0.3151=0.3151\imes100\%=31.51\%.$

Thus, the required probability is 31.51%.