SOLUTION: A jar of alphabet tiles contains 10 unique consonant tiles and 5 unique vowel tiles. If 5 tiles are picked randomly, the probability that 3 are consonants and 2 are vowels is ? .

Question 999165: A jar of alphabet tiles contains 10 unique consonant tiles and 5 unique vowel tiles.
If 5 tiles are picked randomly, the probability that 3 are consonants and 2 are vowels is ?
.

Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

Problem:
Given 15 tiles, with 10 unique consonants, and 5 unique vowels.
Pick 5 randomly (without replacement), find probability of picking 3 consonants and 2 vowels.

Solution:
The simplest way is to use the Hypergeometric distribution, with
N=15 (total number of tiles)
n=5 (number of tiles picked)
k=10 (number of successes, initially)
r=3 (number of successes for which to find),
finally,
C(a,b)=number of combinations of choosing b objects out of a
= a!/(b!(a-b)!)
then
P(r=3)=C(n,r)C(N-n,k-r)/C(N,k)
=C(5,3)C(15-5,10-3)/C(15,10)
=10*120/3003
=400/1001

Another way to approach it would be:
Let
C=event of drawing a consonant
V=event of drawing a vowel
Then
P(CCCVV)=(10/15)(9/14)(8/13)(5/12)(4/11)=40/1001
Since there are C(5,3) ways to arrange 3 consonants and 2 vowels, we multiply
the above by C(5,3)=5!/(3!2!)=10
So the final answer is P(r=3)=40/1001*10=400/1001 as before.
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