Probablity

parveen110 wrote:Veritas prep question.

There are three red balls and two black balls in a jar. What is the probability
that John gets at least one red ball when he picks two balls at random out of
the jar at the same time?

The answer is: 9/10

My approach:

I am considering complementary events of taking no red balls.

I may pick up two balls in 5C2 ways.

and then, first black ball may be picked up in 2C1 ways and the other one in 1C1 ways.

Therefore, (2C1*1C1)/(5C2)=1/5

so, probablity of picking one red ball is: 1-(1/5)=4/5.

Where's my approach flawed?

Dear parveen110,
I'm happy to help.

You may find this blog helpful:
https://magoosh.com/gmat/2014/gmat-advan ... -problems/

Here, the question uses the phrase "at the same time", which I believe is Veritas' way of saying the balls are picked without replacement. In probability, we can never consider two choices truly simultaneous. If one ball is picked from five and not replaced, then the next ball is picked from the remaining four. Your mistake was to treat the problem as if two choices could be truly simultaneous and, therefore, not influence one another.

Another way to say it --- you treated two ways to pick the first black ball, and one way to pick the second, and got 2, but either black ball could be the first, so there is a redundancy here. If we have Black ball P and Black ball Q, we could pick Q then P, or P then Q, and either way, we would wind up with P & Q. Thus, we have to divide by 2 to avoid the redundancy.

I will show two approaches.

Approach #1: individual choices & conditional probability
The probability of picking first ball Black = 2/5
The probability of picking second ball Black, given that first was black = 1/4
(2/5)*(1/4) = 1/10 = probability both are black.

Approach #2: using counting techniques
See: https://magoosh.com/gmat/2013/gmat-proba ... echniques/
Total number of pairs = 5C2 = 10
Total number of pairs of two black balls = 2C2 = 1
Probability of two black balls = 1/10

Either way, we subtract this from 1 to get the OA of 9/10.

Does all this make sense?
Mike