Question

Please show your work. Thanks!

1)In the Science department, suppose there are 3 female professors and and 8 male professors. A committee is to be made of 4 persons. If the committee members are selected at random, what is the probability that there will be the same number of males on the committee as females?
a)0.25
b)0.28
c)0.31
d)0.34
e)0.83

2)If a coin is flipped 3 times, what is the probability of getting exactly 2 heads?

a)0
b)0.1
c)0.125
d)0.25
e)0.375

3)If a coin is flipped 4 times, what is the probability of getting exactly 3 heads, given that at least 1 head is obtained?

a)0.267
b)0.357
c)0.392
d)0.412
e)0.441

Answer 1

1) P(same number of female and male in the committee) = P(2 male and 2 female)

= 8C2 * 3C2 / 11C4

= 0.25

Option-A) 0.25

2) P(head) = 0.5

P(exactly 2 heads) = 3C2 * 0.5² * ( 1 - 0.5)¹ = 0.375

Option-E) 0.375

3) P(X > 1) = 1 - P(X = 0) = 1 - 4C0 * 0.5⁰ * 0.5⁴ = 1 - 0.0625 = 0.9375

P(X = 3) = 4C3 * 0.5³ * 0.5 = 0.25

P(X = 3 | X > 1) = P(X = 3 and X > 1) / P(X > 1)

= P(X = 3) / P(X > 1)

= 0.25 / 0.9375

= 0.267

Option-A) 0.267