Card Probability Problem

Can someone tell me if I am taking the correct approach to this problem?


Given 4 separate decks of 52 cards, what are the odds that the Ace of Hearts will be drawn in the first 4 cards of each separate deck, in 2 out of the 4 decks? Please explain the steps taken to make your determination since we are looking not only for the correct answer, but how you came to you final conclusion.

Probability of getting ace of hearts in 52 card deck = 1/52

If we assume that we take the first 4 cards without looking at them and then flip them over we have 4/52 (1/13) chance of getting ace of hearts right?

So does this mean that if we have 4 situations (separate events) where we need to get that ace of hearts within the first 4 cards of 4 different decks, is it not just taking (1/13) * (1/13) * (1/13) * (1/13) to get 1/28561 ???

2 out of the 4 decks would be half of that probability right? 2/28561 ??

Am I on the right track?

You did a good job on the first one. It is correct.
But the second is binominal:
\(\displaystyle {4 \choose 2} \left( {\frac{4}{{52}}} \right)^2 \left( {\frac{{48}}{{52}}} \right)^2\)

Hmm, can you elaborate?

It has been a while since I did binomials. Do you have a good web reference that I can lookup that shows me how they work?

I am a little confused...

thanks for the reply though!

Binomial: probability of p. Exactly K successes out of independent N trials is:
\(\displaystyle { N \choose K} \left( p \right)^K \left( {1 - p} \right)^{N - k}.\)

this sound ok?

Thank you so much for your help, can you let me know if this looks correct?

Since there is only one ace of hearts in a 52 card deck. The probability of drawing an ace of hearts is 1/52. Now if we draw 4 random cards from a deck of 52, the probability of getting an ace within those 4 cards is 4/52 or 1/13. Now we expand the problem to include 4 separate decks of 52 cards each. Since drawing cards from one deck has no impact on drawing cards from another 3 decks, we consider the events independent. This means that the probability of drawing an ace of hearts in each of the 4 separate decks of 52 cards, drawing 4 cards per deck is (1/13) * (1/13) * (1/13) * (1/13) or (1/28561). Now in the case of drawing an ace of hearts in 2 of the 4 decks, using the same method of 4 cards per deck, we use a binomial method of getting exactly 2 successes out of 4 independent trials:

(4 choose 2)(1/13)^2 * (12/13)^2 = (4!/(2!*2!))*(1/13)^2 * (12/13)^2
= 6 * (1/169) * (144/169)
= 864/28651



thanks again!

YES! That is correct.