Question

Mr. Greenjeans 7th grade agriculture class has 13 boys and 15 girls. If Mr. Greenjeans were to randomly select 7 children from this class, what is the probability that fewer than 3 girls will be selected?

please show me step by step how to figure this out, if there is a way to do it in calculator please show me that way

Answer 1

There are 13 boys and 15 girls, total students = 13+15 = 28

The number of ways of selecting r students from a group of n students = ⁿC_r = n! /((n - r)!r!)

We have to randomly select 7 children from this class which can be done in ²⁸C₇ ways.

Now it is given that to calculate probability that fewer than 3 girls will be selected which means out of 7 children there may be no girls or 1 girl or 2 girls.

P( no girls or 1 girl or 2 girls) = P( no girls) + P(1 girl ) +P(2 girls.)

P(nogirls) = ?

out of 13 boys 7 boys should be selected this can be done in ¹³C₇ ways, 0 girls should be selected this can be done in ¹⁵C₀

P(nogirls) = (¹³C₇ * ¹⁵C₀ )/²⁸C₇ = 1/690

similarly P(1 girl) = (¹³C₆* ¹⁵C₁)/²⁸C₇ = 1/46

P(2girl) = (¹³C₅* ¹⁵C₂)/²⁸C₇ = 21/184

Now probability that fewer than 3 girls = 1/690 + 1/46 + 21/184 = 379/2760 = 0.1373