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

An integer is selected at random from the set of all positive integers less than or equal to 40. Given that the selected integer is a multiple of 3, what is the probability that it is also a multiple of 4?

A
$\frac{3}{40}$
$\frac{3}{13}$ Answer
C
$\frac{3}{10}$
D
$\frac{1}{4}$
E
$\frac{4}{13}$

Answer

The probability that the selected integer is a multiple of 4, given that it is a multiple of 3, is

\frac{3}{13}

The condition 'given that the selected integer is a multiple of 3' restricts the sample space to the 13 positive integers up to 40 that are divisible by 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, and 39. Among these 13 numbers, those that are also multiples of 4 are multiples of 12, namely 12, 24, and 36 (3 numbers). Therefore, the conditional probability is

\frac{3}{13}

Step-by-Step Solution

Identify the restricted sample space defined by the condition.

The positive integers less than or equal to 40 that are multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, and 39. There are 13 such integers.

For conditional probability

P(A \mid B)

, the sample space must be restricted to all outcomes satisfying condition

B

(being a multiple of 3).

Identify the favorable outcomes within the restricted sample space.

Integers that are multiples of both 3 and 4 must be multiples of 12. Among the 13 multiples of 3, those that are also multiples of 12 are: 12, 24, and 36. There are 3 such integers.

The numerator of

P(A \mid B)

counts the elements in the intersection

A \cap B

Calculate the conditional probability.

P(\text{Multiple of 4} \mid \text{Multiple of 3}) = \frac{3}{13}

Divide the number of favorable outcomes (3) by the total number of outcomes in the restricted sample space (13).

Key Concept

Conditional Probability with Restricted Sample Space

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic