probability without replacement

Question

A deck of playing cards has 52 cards. There are 4 jacks and 4 aces. Assuming the jack is not replaced, what is the probability of drawing a jack and then an ace from the deck?

Answer 1

Hi Erin,

The probability of drawing a jack and then an ace is the probability of drawing a jack without replacement multiplied by the probability of drawing an ace right after.

Let J = Event of drawing a jack and

A = Event of drawing an ace after a Jack

Then P(J) = 4/52 = 1/13 (because every card is equally likely to be drawn)

and the probability of A is really a conditional probability. The reason why is because the probability of drawing A is affected by the fact that we drew the previous card without replacement. In particular, it will be

P(A) = P(A|J) = 4/51 (because there are now 51 cards in the deck but still 4 aces. So the answer will be

P = 4/(13*51) = 4/663