A very large student body is 60% female. If 4 students are selected at random, what is the probability

Question

a) They are all females 

b) at least 1 is male 

Answer 1

HI Myra,

 

The probability of drawing a female student, P(F), is 60% or 0.6. Then, the probability of drawing male student, P(M), must be 1 - 0.6 = 0.4. Probabilities of different events must add to 1.

 

a) The probability that all four of the students are females will be

0.6*0.6*0.6*0.6 = 0.1296

The reason that the information "a very large student body.." is given is that when we draw four students, it will have an insignificant impact on the conditional probability. That is the probability of drawing a female student after we already pick one female student will not change much.

 

b) Drawing at least one male student includes the cases of drawing 1, 2, 3 or 4 male students. So, we can simply consider its complement, which is drawing no male students. The probability of drawing no male students is equal to the probability of drawing a female student, which is 0.6. So, the probability of drawing at least 1 male student when 4 students are drawn is equal to 1 - (0.6)⁴ = 0.8704.

If you solve this using binomial probability for considering drawing 1, 2, 3 or 4 male student should yield the same answer

 

Answer 2

P{student is female} = p = 0.6

The probability that r students are female from n selected students is C(n,r)p^r(1-p)^n-r - Just memorize this formula for future uses.

 

For this problem

 

a. The probability that all 4 selected students are female is

P{all 4 students are female} = C(4, 4)(0.6)⁴ = 0.1296

 

b. The probability that at least one student is male

P(at least 1 student is male}

= 1 - P{no student is male}

= 1 - P{all 4 students are female}

= 1- 0.1296

= 0.8704

 

Enjoy!