GRE® Probability Practice

Concept: Reordering of n objects; Quantitative Comparison

This GRE quant practice question is a quantitative comparison question. Comparing probabilities of two events and identifying which between the two has a larger probability. Concept tested: number of ways of reordering elements when all are distinct and when not all are same.

Question: Quantity A Quantity B The probability that a word selected from the set of all rearrangements of the letters of the word "Math" results in "Math" The probability that a word selected from the set of all rearrangements of the letters of the word "Good" results in "Good"

1. Quantity A is greater
2. Quantity B is greater
3. The two quantities are equal
4. The relationship cannot be determined from the information given

Video Explanation

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Use these hints to get the answer

1. Compute the number of ways the letters of the word 'Math' can be reordered.
2. Compute the probability of getting the word 'Math'.
3. Compute the number of ways the letters of the word 'Good' can be reordered.
4. Compute the probability of getting the word 'Good'.

Evaluate Quantity A

Compute the number of ways the letters of the word 'Math' can be reordered.

All 4 letters of the word 'Math' are distinct.

The letters of the word 'Math' can therefore, be reordered in 4! ways = 24 ways.

Compute the required probability

Of the 24 different word orders, only one will be 'Math'.

Therefore, the probability that a word selected out of the 24 turns out to be 'Math' is $\frac{1}{24}$

Evaluate Quantity B

Compute the number of ways the letters of the word 'Good' can be reordered.

Two of the 4 letters of the word 'Good' repeat.

The letters of the word 'Good' can therefore, be reordered in $\frac{4!}{2!}$ = 12 ways.

Compute the required probability

Of the 12 different word orders, only one will be 'Good'.

Therefore, the probability that a word selected out of the 12 turns out to be 'Good' is $\frac{1}{12}$

The Comparison

Quantity A value = $\frac{1}{24}$       quantity B value = $\frac{1}{12}$

Between fractions having the same numerator, the one with the smaller denominator is the larger fraction.

Fraction in quantity B is 1 out 12 parts - while the fraction in quantity A is 1 out of 24 parts - so quantity A is smaller.

Quantity B is greater. Choice B is the answer.