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PS · Problem Solving

655-705

Probability

There are three boys aged $3$ , $5$ , and $8$ , and three girls aged $4$ , $7$ , and $10$ . If one boy and two girls are chosen at random, and the sum of their ages is $z$ , what is the difference between the probability that $z$ is even and the probability that $z$ is odd?

Answer Choices

Correct answer marked below

A

\frac{5}{9}

B

\frac{4}{9}

C

\frac{3}{9}

D

\frac{2}{9}

\frac{1}{9}

Correct

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Explanation

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Official Explanation

Official

Alright, I have this probability problem to solve:

Problem Statement:

There are three boys aged

3

,

5

, and

8

, and three girls aged

4

,

7

, and

10

. If one boy and two girls are chosen at random, and the sum of their ages is

z

, what is the difference between the probability that

z

is even and the probability that

z

is odd?

Understanding the Problem

First, I need to understand what's being asked:

Selection: We need to choose 1 boy out of 3 and 2 girls out of 3.

Sum of Ages: For each combination, we need to find the sum of the ages of the selected boy and two girls.

Probability: We need to find the probability that the sum

z

is even and the probability that it's odd.

Final Answer: The difference between the even probability and the odd probability.

Step 1: List the Ages

First, let's list the ages:

- Boys: 3, 5, 8

- Girls: 4, 7, 10

Step 2: Understand the Selection Process

We need to choose:

1 boy: There are 3 choices here (since there are 3 boys).

2 girls: Since the order doesn't matter, we need combinations. The number of ways to choose 2 girls out of 3 is C(3,2) = 3.

Total number of combinations = number of boys × number of girl pairs = 3 × 3 = 9.

Step 3: Find All Possible Sums

For each combination of 1 boy and 2 girls, we need to find the sum of their ages.

#### Boys:

Boy1: 3 years

Boy2: 5 years

Boy3: 8 years

#### Girl Pairs:

Since the order doesn't matter, the pairs are:

Girl1 and Girl2: 4 + 7 = 11

Girl1 and Girl3: 4 + 10 = 14

Girl2 and Girl3: 7 + 10 = 17

Now, for each boy, we add his age to each of the girl pairs' sums.

Boy1 (3 years):

- Sum with Girl Pair1 (11): 3 + 11 = 14 → even…

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There are three boys aged $3$ , $5$ , and $8$ , and three girls aged $4$ , $7$ , and $10$ . If one boy and two girls are chosen at random, and the sum of their ages is $z$ , what is the difference between the probability that $z$ is even and the probability that $z$ is odd?

Answer Choices

See the full step-by-step explanation

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Official Explanation

Understanding the Problem

Step 1: List the Ages

Step 2: Understand the Selection Process

Step 3: Find All Possible Sums

Read the full explanation

No comments yet

What it tests

Common trap

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There are three boys aged 333, 555, and 888, and three girls aged 444, 777, and 101010. If one boy and two girls are chosen at random, and the sum of their ages is zzz, what is the difference between the probability that zzz is even and the probability that zzz is odd?

Answer Choices

See the full step-by-step explanation

Track your performance and improve

Official Explanation

Understanding the Problem

Step 1: List the Ages

Step 2: Understand the Selection Process

Step 3: Find All Possible Sums

Read the full explanation

No comments yet

What it tests

Common trap

Unlock full explanations

There are three boys aged $3$ , $5$ , and $8$ , and three girls aged $4$ , $7$ , and $10$ . If one boy and two girls are chosen at random, and the sum of their ages is $z$ , what is the difference between the probability that $z$ is even and the probability that $z$ is odd?