An unfair coin with...

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 15 girls. Otherwise, a student from a different class containing 8 boys and 11 girls is selected.

 

What is the probability of selecting a girl?

 

 

What is the probability of selecting a girl, given that the flip resulted in heads?

Answer 1

The second question is easier, so let's get it out of the way first.

 

The total size of the class we choose from when flipping heads is 12+15 = 27 and 15 of them are girls. Therefore

 

Pr[Girl | H] = 15/27 = 5/9

 

Now for the first question. Here you use the Law of Total Probability (LOTP).

 

Start by finding the other conditional probability:

 

Pr[Girl | T] = 11 / (8+11) = 11/19

 

By LOTP:

 

Pr[Girl] = Pr[Girl | H] Pr[H] + Pr[Girl | T] Pr[T]

 

= (5/9)(1/5) + (11/19)(4/5) = 1/9 + 44/95 = 491/855

Answer 2

Well, you want to break this down into parts.

 

1. Find the probability that the coin is heads = .2

2. Find the probability that the coin is tails: remember that the maximum probability is 1, meaning that something is 100% probable and therefore going to happen. Therefore, the probability of the coin landing face up is 1 and if the probability of it being heads is 0.2, then determining the other side is simple.

3. Find the probability of selecting a girl in the first class of 12 boys and 15 girls: determining probability here is about putting the group that you want over the total group

4. Find the probability of selecting a girl from a different class

5. Find the probability of selecting a girl when it is heads: you want to multiply the probability of the coin being heads by the probability of selecting a girl from the respective class.

6. Find the probability of selecting a girl when it is tails

7. Find the probability of selecting a girl in general = (the probability of selecting a girl when it is heads) + (the probability of selecting a girl when it is tails)