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QuestionFind the probability of selecting 2 Democrats and 2 Republicans from a committee of 4 chosen from 8 Democrats and 5 Republicans.

Studdy Solution

STEP 1

Assumptions1. The city council consists of8 Democrats and5 Republicans. . A committee of4 people is to be selected.
3. We need to find the probability of selecting Democrats and Republicans.

STEP 2

First, we need to find the total number of ways to select4 people from the13 members of the city council. This can be done using the combination formulaC(n,k)=n!k!(n−k)!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.

STEP 3

Plug in the given values for n and k to calculate the total number of ways to select people from the13 members of the city council.
C(13,)=13!!(13−)!C(13,) = \frac{13!}{!(13-)!}

STEP 4

Calculate the total number of ways to select4 people from the13 members of the city council.
C(13,4)=13!4!(13−4)!=715C(13,4) = \frac{13!}{4!(13-4)!} =715

STEP 5

Next, we need to find the number of ways to select2 Democrats from the8 Democrats and2 Republicans from the5 Republicans. This can be done using the combination formula twice.

STEP 6

First, calculate the number of ways to select2 Democrats from the8 Democrats.
C(8,2)=8!2!(8−2)!C(8,2) = \frac{8!}{2!(8-2)!}

STEP 7

Calculate the number of ways to select2 Democrats from the Democrats.
C(,2)=!2!(−2)!=28C(,2) = \frac{!}{2!(-2)!} =28

STEP 8

Next, calculate the number of ways to select2 Republicans from the5 Republicans.
C(5,2)=5!2!(5−2)!C(5,2) = \frac{5!}{2!(5-2)!}

STEP 9

Calculate the number of ways to select2 Republicans from the5 Republicans.
C(5,2)=5!2!(5−2)!=C(5,2) = \frac{5!}{2!(5-2)!} =

STEP 10

Now, multiply the number of ways to select2 Democrats and2 Republicans to find the total number of ways to select2 Democrats and2 Republicans.
Total ways=C(8,2)timesC(5,2)Total\, ways = C(8,2) \\times C(5,2)

STEP 11

Plug in the values for C(8,) and C(5,) to calculate the total number of ways to select Democrats and Republicans.
Total ways=28times10Total\, ways =28 \\times10

STEP 12

Calculate the total number of ways to select2 Democrats and2 Republicans.
Total ways=28times10=280Total\, ways =28 \\times10 =280

STEP 13

Finally, we can find the probability of selecting2 Democrats and2 Republicans by dividing the number of ways to select2 Democrats and2 Republicans by the total number of ways to select people.
Probability=Total waysC(13,)Probability = \frac{Total\, ways}{C(13,)}

STEP 14

Plug in the values for Total ways and C(13,4) to calculate the probability.
Probability=280715Probability = \frac{280}{715}The probability of selecting2 Democrats and2 Republicans is 280715\frac{280}{715}.

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