# Probability regarding a batch of 3

#### J_Rod

Joined Nov 4, 2014
109
I don't understand what this probability question is asking.

It states:
"A bowl contains 3 red, 4 blue, and 5 green chips. If 3 chips are simultaneously drawn at random, what is the probability that all 3 chips are the same color? What is the probability that exactly 2 chips are the same color?"

Here is what I worked:
P(3 red) = 3/12 *2/11 *1/10
At first there are 3 red chips, then if one is removed, there are 2, and only 11 chips, and then only 1 red chip and 10 total chips
P(3 green) = 4/12 *3/11 *2/10
(Then similalry for green and blue)
P(3 blue) = 5/12 *4/11 *3/10
Well this method gives 3 answers, but does the question want just one answer? Would I multiply the three probabilities?

P(2 red) = 3/12 *2/11 *9/10
There are 3 red chips to choose, then removing 1 that leaves 2 chips out of 11, but then there are 10 chips, with 9 that are not red, so the probability of not picking another red is 9/10
P(2 green) = 4/12 *3/11 *8/10
Likewise probabilities for green and blue were found
P(2 blue) = 5/12 *4/11 *7/10

#### WBahn

Joined Mar 31, 2012
26,398
It's asking for the probability that all three chips are the same color, regardless of what that color is.

Why would you multiply the probabilities? Wouldn't that make the probability go down? Does that make sense? If I first told you that you would win only if all three chips turn out red, and then I told you that you would win if all three chips turned out red or if all three chips turned out blue, would you really think that your overall odds of winning would go down?

• J_Rod

#### J_Rod

Joined Nov 4, 2014
109
you would win if all three chips turned out red or if all three chips turned out blue,
Your odds of winning would go up. Is it because these are mutually exclusive events I have to add (not multiply) together the probabilities?
There would be 3 possible results: all same color, exactly 2 of same color, or all different colors. The sum of these probabilities is 1, right? Likewise, for the case of exactly 2 chips of the same color, I can add together the probabilities because the events are mutually exclusive?

#### WBahn

Joined Mar 31, 2012
26,398
It sounds like you've basically got it.

The first question is asking for answers to two different questions. The odds of getting all three of the same color is one question, and the odds of getting exactly two of the same color is the second question.

You are correct that you can just add the probabilities of two mutually exclusive events together (and that caveat regarding mutual exclusivity is critical).

• J_Rod