You are dealt five cards from a standard and shuffled deck of playing cards. Note that a standard deck has 52 cards and four of those are kings. What is the probability that you'll have at most three kings in your hand?
I know that the answer is $\frac{54144}{54145}$ from the answer key, and I know that the sample space is ${^{52}\mathrm C_5}$. What I don't get is how to find the event.
Do I just add the combination for each number of kings together (${^5\mathrm C_2} + {^5\mathrm C_3}$)?
Or do I need to multiply as well to account for the other cards in the 5-card hand?
Best Answer
The probability that you will have at most 3 kings is the probability that you will have less than 4.
$$\mathsf P(K\leq 3) = 1 -\mathsf P(K=4)$$
The probability that you will have exactly all four kings is the count of ways to select 4 kings and 1 other card divided by the count of ways to select any 5 cards.
$$\mathsf P(K=4)~=~\dfrac{{^{4}\mathrm C_{4}}\cdot{^{(52-4)}\mathrm C_{(5-4)}}}{^{52}\mathrm C_5}$$
Put it together.
$$\mathsf P(K\leq 3) ~=~ 1 -\dfrac{{^{4}\mathrm C_{4}}\cdot{^{(52-4)}\mathrm C_{(5-4)}}}{^{52}\mathrm C_5}$$