Question

Beyonc´e is having twins! Twins come in two different types: identical or fraternal. Identical twins are alike genetically (including gender) and are typically very similar in appearance. Fraternal twins are genetically distinct and are not necessarily of the same gender. Suppose there is a 30% chance the Beyonc´e will have identical twins and that each child is a boy or a girl with equal probability, independently.a. Find the probability that Beyonce has one boy and one girl. b. Given that Beyonce has two boys, what is the probability that they are fraternal twins.

Answer 1

0.35 or 35%

0.5385 or 53.85%

Explanation:

a. If Beyoncé is having a girl and a boy, then the twins cannot be identical, the probability that they are not identical is 70%. The sample space for the genders of non identical twins is:

S={BB, BG, GB, GG}

Thus, there is a 50% chance that the twins are a boy and a girl.

The probability that Beyoncé has one boy and one girl is:

`P=0.7*0.5 =0.35`

The probability is 0.35 or 35%.

b. The probability that the twins are fraternal given that they are two boys is determined as the probability of two fraternal boys being born divided by the probability of two boys being born.

For fraternal twins, there is a 25% chance that both are boys:

`P(F\ \&\ BB)=0.7*0.25=0.175`

If they are identical, there is a 50% chance that both are boys. The probability that they are both boys is:

`P(BB) = P(I\ \&\ BB)+P(F\ \&\ BB)=0.3*0.5+0.7*0.25=0.325`

Therefore:

`P(F|BB)=( P(F\ \&\ BB))/( P(I\ \&\ BB)+P(F\ \&\ BB)) \\P(F|BB)=(0.175)/(0.325) =0.5385`

Given that Beyoncé has two boys, the probability that they are fraternal twins. is 0.5385.