Question

select the correct answer.
the chaperones for a field trip to the museum will be 7 history teachers and 13 science teachers. each student on the trip will be assigned to a group led by two chaperones.
what is the approximate probability that a random student on the trip will be assigned to a group led by two history teachers?
a. 12.3%
b. 2.3%
c. 11.1%
d. 28.6%

Explanation:

Step1: Calculate the total number of teachers

The total number of teachers is \(7 + 13=20\) teachers.

Step2: Calculate the number of ways to choose 2 teachers out of 20

The number of ways to choose 2 teachers out of \(n = 20\) is given by the combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\), where \(n=20\) and \(k = 2\). So \(C(20,2)=\frac{20!}{2!(20-2)!}=\frac{20\times19\times18!}{2\times1\times18!}=190\)

Step3: Calculate the number of ways to choose 2 history teachers out of 7

Using the combination formula with \(n = 7\) and \(k=2\), \(C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7\times6\times5!}{2\times1\times5!}=21\)

Step4: Calculate the probability

The probability \(P\) is the number of favorable outcomes (choosing 2 history teachers) divided by the number of total outcomes (choosing any 2 teachers). So \(P=\frac{C(7,2)}{C(20,2)}=\frac{21}{190}\approx0.1105\approx11.1\%\)

Answer:

C. \(11.1\%\)