Hi, there--

YOUR PROBLEM:
A bag contains 4 red, 5 white and 7 black balls. If 2 balls are taken random, what is the 
probability that both of them will be black or both will be white?

SOLUTION:

First, find the probability that two balls will be black. There are 16 balls altogether. Of these, 
7 of them are black and 9 are not black. Count the possible ways of choosing two black balls 
and 0 non-black balls from the bag (combinations because order doesn't matter):

There are 7C2 ways to choose two black balls = 21
There are 9C0 ways to choose zero non-black balls = 1
There are 16C2 ways to draw any two balls = 120

The probability of drawing two black balls is (7C2)(9C0)/(16C2) = (21)(1)/120 = 21/120

Second, find the probability that both balls will be white. There are 16 balls altogether. Of 
these, 5 are white and 11 are not white. Count the possible ways of choosing two white balls
and 0 non-white balls from the bag:

There are 5C2 ways to choose two white balls = 10
There are 11C0 ways to choose zero non-white balls = 1
There are 16C2 ways to draw any two balls = 120

The probability of drawing two white balls is (5C2)(11C0)/(16C2) = (10)(1)/120 = 10/120

We want the probability that you will draw two black OR two white, so we add these probabilities.

21/120 + 10/120 = 31/120

I hope this helps! Feel free to emil me if you have questions about this explanation.

Mrs. Figgy
math.in.the.vortex@gmail.com