I'm looking into two methods to solve the question of the probability of being dealt exactly two jacks in a 5-card hand. Both make perfect sense to me but give different answers.
Method 1: (4C2)(48C3)/(52C5) = 0.03993
This is just (# favorable outcomes)/(# total outcomes)
This is the (probability of drawing a jack on the first card) * (prob of drawing a jack on the 2nd card) * (prob of drawing a non-jack on the 3rd) * (prob of drawing a non-jack on the 4th) * (prob of drawing a non-jack on the 5th)
Why do these answers differ by a factor of 10? Where am I going wrong?
think very carefully about the language of your question:
"the probability of being dealt exactly two jacks in a 5-card hand."
note that in method 2 you have specified that the J's are in slots 1 and 2 (permutation), whereas in the original question you've asked for the probability of 2 Jacks in a 5 card hand meaning the jacks can be in any of the 5 slots (combination). the factor of 10 in method 1 accounts for the J's being in any position, b/c you have taken taken the 5 cards and selected 2 jacks rather than assigning them fixed positions
...and if you are specifically asking for precisely a pair of jacks, the calculation should be (4C2)(12C3)43/(52C5).
Why is this?
There are 4C2 ways to pick a pair of jacks and among the 48 remaining non-jack cards, there are 48C3 ways to pick the remaining 3 cards of the hand, so I had 4C2*48C3. Does that not work?
There are 4C2 ways to pick a pair of jacks and among the 48 remaining non-jack cards, there are 48C3 ways to pick the remaining 3 cards of the hand, so I had 4C2*48C3. Does that not work?
Your way includes hands with another pair besides the jacks, or even a full house. Bigpooch's formula ensures that the 3 non-jacks are 3 different ranks.
BTW, I mean 2 jacks and 3 non-jack cards; boats and two-pair hands are still considered favourable outcomes
Quote:
Originally Posted by BruceZ
Your way includes hands with another pair besides the jacks, or even a full house. Bigpooch's formula ensures that the 3 non-jacks are 3 different ranks.
Tried to edit this to put it in the OP, but it was after the editing time.
