# Conditional Probability

## Main Question or Discussion Point

I could have posted this in the homework section but I didn't see a section on probability. Also, this is more of a general conditional probability question asked through an example.

A band called Radiohead is inspired by an old band called The Beatles.
50% of music critics think the beatles was a great (G) band, 40% that it
was moderate (M) and 10% that it was awful (A). These critics have also
compiled the following table:

0.8 0.1 0.1
0.1 0.9 0
0.2 0.3 0.5

The table says that the probability of a new band (B2) being great given
that the inspiring band (B1) was great is P (B2 = GjB1 = G) = 0:8: Similarly, P (B2 = GjB1 = M) = 0:1, P (B2 = MjB1 = A) = 0:3, and so on.
Note that the numbers in the rows add up to 1, so the table is a probability
transition matrix.
1. Given what the critics think of the Beatles and the fact that the Beatles
band?
2. What is P (B1 = GjB2 = G)?

For 1 I get 0.50*0.8 + 0.40*0.1 + 0.10*0.2 = 0.46 which I think is correct.

However, I am having trouble with part 2. I can write

P(B1 = G | B2 = G) = P(B2 = G | B1 = G)P(B1 = G) / P(B2 = G)

which leads to 0.8 * P(B1 = G) / P(B2 = G)

I also notice that P(B2 = G) = 1 cause we are told.

P(B1 = G | B2 = G) = 0.8*P(B1 = G)

Is there any way to get an exact answer for this? Or do I have to find a different way to tackle this problem?