A box contains red, blue, and yellow game tokens. The probability of randomly selecting a … — ACT

A box contains red, blue, and yellow game tokens. The probability of randomly selecting a red token is $\frac{3}{8}$ , and the probability of randomly selecting a blue token is $\frac{1}{3}$ . If the box contains exactly $14$ yellow tokens, how many total tokens are in the box?

$48$ Answer
B
$24$
C
$22$
D
$16$
E
$72$

Answer

There are 48 total tokens in the box.

The total probability of all outcomes in the box must equal

1

. Combining the given probabilities for red and blue tokens yields

\frac{3}{8} + \frac{1}{3} = \frac{9}{24} + \frac{8}{24} = \frac{17}{24}

. The remaining probability representing the yellow tokens is

1 - \frac{17}{24} = \frac{7}{24}

. Since the box contains

14

yellow tokens, setting

\frac{7}{24}

of the total number of tokens equal to

14

gives

\frac{7}{24} N = 14

, which simplifies to

N = 48

Step-by-Step Solution

Find the probability of selecting a red or blue token.

\frac{3}{8} + \frac{1}{3} = \frac{9}{24} + \frac{8}{24} = \frac{17}{24}

Since selecting a red token and a blue token are mutually exclusive events, add their probabilities using a common denominator of

24

Calculate the probability of selecting a yellow token.

1 - \frac{17}{24} = \frac{7}{24}

The sum of the probabilities of all possible outcomes (red, blue, yellow) must equal

1

Set up and solve the equation for the total number of tokens

N

\frac{7}{24} N = 14 \implies N = 14 \times \frac{24}{7} = 2 \times 24 = 48

Multiply the count of yellow tokens (

14

) by the reciprocal of the yellow token probability (

\frac{24}{7}

Key Concept

Determining total sample size using complementary probabilities of mutually exclusive events

Estimated Time:1m 15s

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