What is the probability of getting exactly 3 kings when drawing 5 cards from a deck of 52 cards?

K K K Not K Not K

4/52 x 3/51 x 2/50 x 48/49 x 47/48

I got 47/270725. Is this correct?

Assuming sampling

**without replacement**, no, this answer is not correct since there are actually several different ways you could sample to obtain 3 kings from drawing 5 cards. The one you provided is actually only

**one** of these possible sequences. For example, another possibility is \((K,K,N,K,N)\).

Essentially, you need to find all the possible combinations where 3 kings are obtained, find their individual probabilities, and then

**add** them all up. To help, you could draw a tree diagram (although it would get messy). Here are some combinations to get you started: \[\text{Pr}(K,K,K,N,N)=\dots\\

\text{Pr}(K,K,N,K,N)=\dots\\

\text{Pr}(K,K,N,N,K)=\dots\\

\vdots\ \ \ \text{etc.}\]

Another way you could answer the problem, which isn't strictly in the study design is to use the

*hypergeometric distribution*, which gives \[\text{Pr}(\text{event})=\frac{\displaystyle\binom{4}{3}\binom{48}{2}}{\displaystyle\binom{52}{5}}=\frac{4\times 1128}{2\,598\,960}=\frac{94}{54145}\]