# Math Help - Card probability question

1. ## Card probability question

Hey everyone, a co-worker and I were playing a board game and afterward were trying to calculate the probability of drawing cards. Neither of us could come up with an answer and so I thought I'd ask here before I go mad trying to puzzle it out.

The way it works is this. There are 29 cards in the deck. 27 of them are unique (let's pretend they're marked 1 thru 27. And the other 2 cards are marked "A". So they match each other but none of the others. We were simply trying to figure out if you were to draw 5 cards at random from the deck. What are the odds that you'll draw the 2 "A" cards?

If anyone can help with the math on this one, my sanity would greatly appreciate it.

2. Hello, Allan!

I simplified the wording of the problem.

There are 29 cards in the deck: 2 A's and 27 Others.

If we draw 5 cards at random from the deck, what is the probability of getting the 2 A's?
There are: . $_{29}C_5 \:=\:{29\choose5} \:=\:\frac{29!}{5!\,24!} \:=\:118,\!755$ possible hands.

We draw 5 cards and want: two A's and three Others.
. . There is: . ${2\choose2} \:=\:1$ way to get two A's.
. . . There are: . ${27\choose3} \:=\:2,\!925$ ways to get three Others.
. . . . . . Hence, there are $2,\!925$ ways to get two A's.

Therefore: . $P(\text{two A's}) \;=\;\frac{2,\!925}{118,\!755} \;=\;\frac{5}{203}$

3. Thanks! I have a quick follow up question.

There are 30 cards in the deck: 3 A's and 27 Others.

If we draw 5 cards at random from the deck, what is the probability of getting any of the 2 A's?