# GMAT Sample Problem: Probability Problem-Solving

by on October 21st, 2010

Try your hand at this sample problem focusing on a specific probability situation.

## Sample Problem:

Each person in Room A is a student, and 1/6 of the students in Room A are seniors.  Each person in Room B is a student, and 5/7 of the students in Room B are seniors.  If 1 student is chosen at random from Room A and 1 student is chosen at random from Room B, what is the probability that exactly 1 of the students chosen is a senior?

(A) 5/42
(B) 37/84
(C) 9/14
(D) 16/21
(E) 37/42

## Solution:

In this problem we are asked to determine the probability of choosing exactly 1 senior.  To be more specific, this means that we will need to select 1 senior and 1 non-senior.

There are two ways this can be done.  We can select a senior from Room A and a non-senior from Room B or we can select a non-senior from Room A and a senior from Room B.  In probability, ‘and’ means multiply while ‘or’ means add.  In this case, we see 2 ‘ands’ indicating that we will multiply probabilities together and one ‘or’ telling us that we will add the resulting products.

Our first option is to get a senior from Room A, which has a probability of 1/6, and a non-senior from Room B, which has a probability of 2/7 (we found this number by taking the probability of selecting a senior in Room B and subtracting it from 1.)  Because this is an ‘and’ situation, we multiply 1/6 * 2/7, which equals 2/42.

For our other option we must find the probability of selecting a non-senior from Room A, which has a probability of 5/6, and senior from Room B, which has a probability of 5/7.  Again, we have an ‘and’ statement, so we multiply.  5/6 * 5/7 = 25/42.

Lastly, we must address the ‘or’ in our original scenario by adding the two probabilities we determined together.  This gives us 2/42 + 25/42 = 27/42.  This simplifies to 9/14, which is choice (C).

• I have a Doubt here - Dont we have to consider the fact that both the students selected from Room A and Room B could be seniors ?

• Hi Anil,
Since the question asked us for "exactly 1" of the students to be a senior, that is what we focused on in solving, just that scenario.

• A good follow up question: What would be the probability of AT LEAST 1 student being a senior?

• At least one senior:

Total number of choices: 6*7=42
Total number of choices where NO senior is chosen: 5*2=10
42-10=32
Probability that at least one senior is chosen:
32/42 = 16/21

• scuse me jackhero

why the second statement is 5*2........??? sorry

• Not choosing a senior in room a is 5/6 and room b 2/7. I guess he multiplied the numerators but I am not sure why.