An integer n is chosen at random from the set of all positive integers less than or equal … — GMAT

An integer $n$ is chosen at random from the set of all positive integers less than or equal to $120$ . What is the probability that $n$ is a multiple of either $3$ or $5$ , but not a multiple of $15$ ?

Answer: 0.4

Answer

The probability that the selected integer is a multiple of either 3 or 5, but not 15, is 0.4 (or 2/5).

The total number of integers from 1 to 120 is 120. Multiples of 3 up to 120 total 40, multiples of 5 total 24, and multiples of 15 total 8. Integers that are multiples of 3 but not 15 number 40 - 8 = 32. Integers that are multiples of 5 but not 15 number 24 - 8 = 16. The total number of favorable outcomes is 32 + 16 = 48. Thus, the probability is 48/120 = 2/5 = 0.4.

Step-by-Step Solution

Count total possible outcomes in the sample space.

The total number of integers from 1 to 120 is 120.

Each integer in the set {1, 2, ..., 120} is equally likely to be selected.

Count the number of multiples of 3, 5, and 15 within the range.

Multiples of 3: 40; Multiples of 5: 24; Multiples of 15: 8.

Since 120 is divisible by 3, 5, and 15, the count of multiples of k up to 120 is 120/k.

Calculate the number of integers that are multiples of 3 or 5, but not 15.

Number of favorable outcomes = (Multiples of 3 only) + (Multiples of 5 only) = (40 - 8) + (24 - 8) = 32 + 16 = 48.

Multiples of 15 are common multiples of both 3 and 5 and must be excluded completely according to the condition 'not a multiple of 15'.

Compute the single-event probability.

Probability = 48 / 120 = 2 / 5 = 0.4.

Probability of a single event is defined as the ratio of favorable outcomes to total possible outcomes.

Key Concept

Basic Single-Event Probability with Set Restrictions

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