Unfair coin, probability of selecting a girl?

Question

An unfair coin with Pr[H]=0.2 is flipped. If the flip results in heads, a student is selected at random from a class of 12 boys and 10 girls. Otherwise, a student from a different class containing 12 boys and 9 girls is selected.

 

What is the probability of selecting a girl?

Answer 1

Ignore the coin flip for now.

 

Let group A be the group with 12 boys and 10 girls.

Let group B be the group with 12 boys and 9 girls.

 

The probability of picking a single girl from group A is 10/22

The probability of picking a single girl from group B is 9/21

 

Now that we know the probability for each group independently, we can add in the coin flip.

 

We know that the probability of landing heads is 2/10, which means there is a 2/10 chance that group A will get picked. Inversely, there is a 8/10 chance of tails, which means that group B will get picked.

 

Therefore we can make the following equation:

 

Chance of girl being picked = (2/10) * (10/22) + (8/10) * (9/21)

= 20/220 + 72/210 ≈ 0.43

 

Therefore, there is a 43% chance of picking a girl.