A student council committee consists of 15 members: 8 juniors and 7 seniors. Among the jun… — ACT

A student council committee consists of $15$ members: $8$ juniors and $7$ seniors. Among the juniors, $3$ are on the debate team. Among the seniors, $4$ are on the debate team. If one committee member is selected at random, what is the probability that the selected member is a senior or is on the debate team?

A
$\frac{4}{15}$
B
$\frac{7}{15}$
$\frac{2}{3}$ Answer
D
$\frac{11}{15}$
E
$\frac{14}{15}$

Answer

The probability that the selected member is a senior or on the debate team is

\frac{2}{3}

The total number of committee members is

15

. The event consists of selecting someone who is either a senior or on the debate team. There are

7

total seniors and

7

total debate members (

3

juniors and

4

seniors). Using the principle of inclusion-exclusion, the number of favorable outcomes is

7 + 7 - 4 = 10

. The probability is

\frac{10}{15}

, which simplifies to

\frac{2}{3}

Step-by-Step Solution

Identify the total number of members in the sample space.

The total number of members is

15

This is given as the total committee size.

Determine the number of favorable outcomes for the event.

Number of seniors =

7

. Number of juniors on the debate team =

3

. Total favorable outcomes =

7 + 3 = 10

To find members who are a senior OR on the debate team, count all seniors (

7

) plus non-senior debate members (

3

) to avoid double-counting.

Calculate and simplify the probability fraction.

\text{Probability} = \frac{10}{15} = \frac{2}{3}

Divide the favorable outcomes (

10

) by total outcomes (

15

) and simplify by dividing numerator and denominator by

5

Key Concept

Probability of Compound Events (Inclusion-Exclusion Principle)

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