Expert: Paul Klarreich - 9/26/2009

Question
Three cards are drawn from a standard deck of 52 cards.
a. What is the expected number of aces?
b. What is the expected number of clubs?


Answer
Questioner: John
Country: United States
Category: Advanced Math
Private: Yes
Subject: probability
Question: Three cards are drawn from a standard deck of 52 cards.
a. What is the expected number of aces?
b. What is the expected number of clubs?
.................................................
Expectation = value of event * p(event)

To get exactly  1 ace:  

number of subsets of 3 cards containing one ace =
   C(4,1) * C(48,2) = 4 * 48 * 47/(2 * 1) = 2 * 48 * 47 -- multiply by 1

number of subsets of 3 cards containing two aces =
   C(4,2) * C(48,1) = (4 3 48)/(2 1) = 2 3 48  -- multiply by 2

number of subsets of 3 cards containing three aces =
   C(4,3) * C(48,0) = 4 3 2/3 2 1 * 1 = 4  -- multiply by 3

Add those 3 results.  That is your NUMERATOR.

number of 3-card subsets:  C(52,3) = 52 51 50/3 2 1. That is your DENOMINATOR.

and you have your expectation.

I think it is:

= 0.22597285067873303167420814479638
...........................................
The clubs problem works the same way, except your first C is C(13,1-2-3))  and
your second C is  C(39,2-1-0))