From a group of 4 women and 7 ​men, a delegation of 3 is selected. Find the expected number of women in the delegation.

Question

What is the expected number of women? (round to two decimal places as needed)

Answer 1

(4)/(4+7) x 3

= 4/11 x 3

E(w) = 12/11 = 1 1/11th women

= 1.090909... of a woman

= 1.09 rounded to 2 decimal places

the expected value is Not an actual value you'd expect

similar to the average family has 2.3 children

"expected value" is just the average

Answer 2

The total number of ways to pick a delegation of 3 from 11 people is ₁₁C₃ = 165.

 

There are four outcomes with respect to how many women are picked, and we can find how many ways each outcome can occur..

 

0 woman: ₄C₀ x ₁₁C₃ = 1 x 11!/(3!x8!) = 35

1 woman: ₄C₁ x ₁₁C₂ = 4!/3! x 11!/(3!x8!) = 4 x 21 = 84

2 women: ₄C₂ x ₁₁C₁ = 4!/(2!x2!) x 11!/(3!x8!) = 6 x 7 = 42

3 women: ₄C₃ x ₁₁C₀ = 4!/3! x 11!/(3!x8!) = 4 x 1 = 4

 

Therefore, the probability of each outcome is as follows.

 

0 woman: 35 / 165
1 woman: 84 / 165
2 women: 42 / 165
3 women: 4 / 165

 

Hence, the expected number of women is

 

0 x 35 / 165 + 1 x 84 / 165 + 2 x 42 / 165 + 3 x 4 / 165

= 0 + 84/165 + 84/165 + 12/165

= 190/165

= 38/33 ≈ 1.15