Question

Suppose that a committee of 12 people is selected in a random manner from a group of 100 people. Determine the probability that two particular people A and B will both be selected.

Answer 1

Total number of people = (n) = 100

number of selection (r) = 12

total number of ways of selecting 12 people out of 100 can be solved using the combination formula C(n,r)

setting n = 100 and r = 12

we get, total number of selecting 12 people = C(100,12)

Now, let us consider the case where two people A and B must be selected in the 12 people committee

So, 2 positions for A and B must be filled, now we can select only 10 people for committee

So, number of ways of selecting 12 people including both A and B = C(n,r)

setting n = 98 and r= 10 (because we have only 98 people after selecting A and B)

we get, number of ways = C(98,10)

Therefore, required probability = (number of ways of selecting 12 people including both A and B)/(total number of ways of selecting 12 people out of 100)

= C(98,10)/C(100,12)

= [98!/((98-10)!*10!)]/[100!/((100-12)!*12!)]

= [98!/(88!*10!)]/[100!/(88!*12!)]

= 1/75

= 0.0133