From a pack of 52 cards, two cards are drawn randomly. What is the probability of both the cards being kings?

Question

How do you do this? So, I was doing a mini maths test and the question popped up. Because there were 4 kings, so I did 4/52(there were 52 cards) and then simplified it. I got it wrong. So confused! (No replacement, but it will be great if you guys show me with replacement as well).

Answer 1

I will give another way to think about this problem, which might help the OP.

The sample space in this problem is ${52}\choose{2}$ since there are ${52}\choose{2}$ ways to pick a pair of cards. There are 4 kings, therefore there are ${4}\choose{2}$ ways of picking a pair of kings. From here we see that the probability of getting 2 kings is

$$P(\textrm{getting 2 kings }) = \frac{\textrm{number of ways the desired event can occur}}{\textrm{number of possible outcomes}}=\frac{{4}\choose{2}}{{52}\choose{2}}$$

Answer 2

Hint: this can be solved using a “probability tree”

• start by drawing the first card. What are the odds of it being a king? Simply $4/52$.
• Now that you were lucky enough to draw the first king there are only ... kings left in the deck of only ... cards.
• Conclude.