Question

Read the following passage:

Recent studies suggest the roughly 12% of the world population is left-handed. Depending upon the parents, the chances of having a left-handed child are as follows: A :  When both father and mother are left-handed: Chances of left-handed child is 24%. B :  When father is right-handed and mother is left-handed: Chances of left-handed child is 22%. C :  When father is left-handed and mother is right-handed: Chances of left-handed child is 17%. D :  When both father and mother are right-handed: Chances of left-handed child is 9%. Assuming that P(A) = P(B) = P(C) = P(D) = `1/4` and L denotes the event that child is left-handed.

Based on the above information, answer the following questions:

1. Find `P(L/C)` (1)
2. Find `P(overlineL/A)` (1)
3. (a) Find `P(A/L)` (2)
   OR
   (b) Find the probability that a randomly selected child is left-handed given that exactly one of the parents is left-handed. (2)

Answer 1

Given, P(L) = `12/100`

`P(L) P(overlineL) = 1 - 12/100 = 88/100`

and P(A) = P(B) = P(C) = P(D) = `1/4`

`P(L/A) = 24/100`,

`P(L/B) = 22/100`,

`P(L/C) = 17/100`,

`P(L/D) = 9/100`

(i) `P(L/C) = 17/100`, from the given data.

(ii) `P(overlineL/A) = (P(overlineL ∩ A))/(P(A))`

= `(P(A) - P(L ∩ A))/(P(A))`

= `1 - (P(L ∩ A))/(P(A))`

= `1 - P(L/A)`

= `1 - 24/100`

= `(100 - 24)/100`

= `76/100`

= `38/50`

= `19/25`

(iii) (a) `P(A/L) = (P(A ∩ L))/(P(L))`

But `P(L/A) = (P(A ∩ L))/(P(A))`

`24/100 = (P(A ∩ L))/(1/4)`

`\implies` P(A ∩ L) = `24/100 xx 1/4 = 6/100 = 3/50`

∴ `P(A/L) = (3/50)/(12/100) = (3 xx 100)/(12 xx 50) = 1/2`.

OR

(b) `P(L/(B ∪ C)) = (P[(L) ∩ (B ∪ C)])/(P(B ∪ C))`

= `(P[(L ∩ B) ∪ (L ∩ C)])/(P(B ∪ C))`

= `(P(L ∩ B) + P(L ∩ C) - P(L ∩ B)P(L ∩ C))/(P(B) + P(C) - P(B)P(C))`  ...(As they are independent)

= `(22/100 xx 1/4 + 17/100 xx 1/4 - 22/400 xx 17/400)/(1/4 + 1/4 - 1/4 xx 1/4)`

= `(22/400 + 17/400 - (22 xx 17)/(400 xx 400))/(1/2 - 1/16)`

= `((39/400 - 374/160000))/(1/2 - 1/16)`

= `16/7((39 xx 400 - 374)/160000)`

= `(16 xx 15226)/(7 xx 160000)`

= 0.217

= 0.22