Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$Question 12, 11.5.13 HW Score: 47.62%,10 of 21 points Points: 0 of 1 Save ct: 1 A city council consists of six Democrats and five Republicans. If a committee of seven people is selected, find the probability of selecting four Democrats and three Republicans. tion 10 (Type a fraction. Simplify your answer.)$

Question $12$ , $11$ . $5$ . $13$ $\newline$ HW Score: $47.62 \%, 10$ of $21$ $\newline$ points $\newline$ Points: $0$ of $1$ $\newline$ Save $\newline$ ct: $1$ $\newline$ A city council consists of six Democrats and five Republicans. If a committee of seven people is selected, find the probability of selecting four Democrats and three Republicans. $\newline$ tion $10$ $\newline$ (Type a fraction. Simplify your answer.)

View step-by-step help

Home

Math Problems

Algebra 1

Probability of independent and dependent events

Full solution

Q. Question

12

11

5

13

\newline

HW Score:

47.62 \%, 10

21

\newline

points

\newline

Points:

0

1

\newline

Save

\newline

ct:

1

\newline

A city council consists of six Democrats and five Republicans. If a committee of seven people is selected, find the probability of selecting four Democrats and three Republicans.

\newline

tion

10

\newline

(Type a fraction. Simplify your answer.)

Calculate Total Ways: Calculate the total number of ways to select a committee of seven people from the eleven council members without regard to party affiliation. $\newline$ The total number of ways to select seven people from eleven is given by the combination formula $C(n, k) = \frac{n!}{k!(n - k)!}$ , where $n$ is the total number of items, $k$ is the number of items to choose, and “!” denotes factorial. $\newline$ $C(11, 7) = \frac{11!}{7!(11 - 7)!} = \frac{11!}{7!4!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330$ .
Select Democrats: Calculate the number of ways to select four Democrats from the six available. $\newline$ $C(6, 4) = \frac{6!}{4!(6 - 4)!} = \frac{6!}{4!2!} = \frac{6 \times 5}{2 \times 1} = 15$ .
Select Republicans: Calculate the number of ways to select three Republicans from the five available. $\newline$ $C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3!2!} = \frac{(5 \times 4)}{(2 \times 1)} = 10$ .
Calculate Probability: Calculate the probability of selecting four Democrats and three Republicans. $\newline$ The probability is the number of favorable outcomes divided by the total number of outcomes. $\newline$ The number of favorable outcomes is the product of the number of ways to select four Democrats and the number of ways to select three Republicans. $\newline$ Number of favorable outcomes = $C(6, 4) \times C(5, 3) = 15 \times 10 = 150$ .
Calculate Final Probability: Calculate the final probability as a fraction. $\newline$ Probability = Number of favorable outcomes / Total number of outcomes = $\frac{150}{330}$ . $\newline$ This fraction can be simplified by dividing both the numerator and the denominator by $10$ . $\newline$ Probability = $\left(\frac{150}{10}\right) / \left(\frac{330}{10}\right) = \frac{15}{33}$ . $\newline$ Further simplification by dividing both numerator and denominator by $3$ gives us: $\newline$ Probability = $\left(\frac{15}{3}\right) / \left(\frac{33}{3}\right) = \frac{5}{11}$ .