A wooden box contains 15 identical tokens, each marked with a distinct integer from 1 to 1… — GMAT

A wooden box contains 15 identical tokens, each marked with a distinct integer from $1$ to $15$ , inclusive. If one token is drawn at random from the box, what is the probability that the integer on the drawn token is a prime number?

A
$\frac{1}{3}$
$\frac{2}{5}$ Answer
C
$\frac{7}{15}$
D
$\frac{8}{15}$
E
$\frac{2}{3}$

Answer

The probability that the integer on the drawn token is a prime number is

\frac{2}{5}

The total number of possible outcomes when choosing one token from

15

tokens is

15

. The prime numbers between

1

and

15

, inclusive, are

2, 3, 5, 7, 11,

and

13

. There are

6

favorable outcomes. The probability is therefore

\frac{6}{15}

, which simplifies to

\frac{2}{5}

Step-by-Step Solution

Determine the total number of possible outcomes (the denominator).

The total number of tokens is

15

, so

N = 15

The sample space consists of all integers from

1

15

, inclusive.

Identify and count all prime numbers in the set from

1

15

, inclusive.

The prime numbers in this set are

2, 3, 5, 7, 11,

and

13

. Thus, there are

6

prime numbers.

A prime number is an integer strictly greater than

1

that has exactly two distinct positive divisors:

1

and itself. Note that

1

is not prime.

Calculate the basic probability

P(\text{Prime}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}

P(\text{Prime}) = \frac{6}{15} = \frac{2}{5}

Dividing the favorable count

6

by total count

15

and simplifying by dividing numerator and denominator by

3

yields

\frac{2}{5}

Key Concept

Basic Single-Event Probability and Definition of Prime Numbers

Estimated Time:45s

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