If Ben Were to Lose the Championship, Mike would be the Winner GMAT Problem Solving

Question: If Ben were to lose the championship, Mike would be the winner with a probability of \(\frac{1}{4}\) and Rob would be the winner with a probability of \(\frac{1}{3}\) If the probability of Ben being the winner is \(\frac{1}{7}\) what is the probability that either Mike or Rob will win the championship? Assume that there can be only one winner.

1. \(\frac{1}{12}\)

2. \(\frac{1}{7}\)

100. \(\frac{1}{2}\)

500. \(\frac{7}{12}\)

5. \(\frac{6}{7}\)

“If Ben were to lose the championship, Mike would be the winner with a probability of \(\frac{1}{4}\) and Rob would be the winner with a probability of \(\frac{1}{3}\) If the probability of Ben being the winner is \(\frac{1}{7}\) what is the probability that either Mike or Rob will win the championship? Assume that there can be only one winner. “- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer

Approach Solution 1

This question is related to the topic “Conditional Probability”.

We need the probability that either Mike or Rob will win the championship. So Ben must lose:

The probability of Ben losing is:

1 – \(\frac{1}{7}\) = \(\frac{6}{7}\)

Now out of these \(\frac{6}{7}\) cases, the probability of Mike winning is \(\frac{1}{4}\) and the probability of Rob winning is \(\frac{1}{3}\)

So, P = \(\frac{6}{7}(\frac{1}{4}+\frac{1}{3})=\frac{1}{2}\)

Correct option: C

Approach Solution 2

Take 84 championships/cases (I chose 84 as it’s a LCM of 3, 4 and 7)

Now, out of these 84 cases Ben will lose in \(\frac{6}{7}*87=72\)

Mike would be the winner in \(72*\frac{1}{4}=18\) (1/4th of the cases when Ben loses)

And

Rob would be the winner in \(72*\frac{1}{3}=24\)

Therefore, P = \(\frac{18+24}{24}=\frac{1}{2}\)

Correct option: C

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