probability trees for guilt versus innocence

[Laura Graham]

For the life of me I can't figure out how to draw the tree for this problem:

Sheep rustling is a crime and a tribunal hears the cases of men accused of it. Three judges hear the case, then vote either I for innocent or G for guilty. The verdict is what at least 2 out of 3 judges decide. The judges vote independently. Judge 1 is correct 1/2 of the time, Judge 2 is correct 2/3 of the time, and Judge 3 is correct 3/4 of the time. By correct we mean that the judge deems a man guilty only if he is guilty. Only 3/4 of the people accused of sheep rustling have actually committed the crime. Draw a probability tree and answer:

a.) If a man is innocent, what is the probability that all 3 judges will find him innocent?

b.) If a man is innocent, what is the probability that at least two out of three judges will find him innocent?

c.) If a man is guilty, what is probability that at least two out of three judges will find him innocent?

d.) If a case is picked at random, what is probability that a verdict of innocent will be returned?

e.) If a man has been declared innocent by the tribunal, what is the probability that he is really innocent?

~My teacher said that the tree starts off with guilty and innocent. I don't understand why you wouldn't start off with the three judges instead. So far on the tree i have only gotten to the 2nd judge. Sorry about my tree-drawing skills. I am about as good with computers as I am with math!

                               -----(G) 3/4
                   -----(G) 2/3
       -----(G) 1/2            -----(I) 1/4
(G) 3/4
       -----(I) 1/2            -----(?)
                   -----(I) 1/3
                               -----(?)


                               -----(?)
                   -----(G) 1/3
       -----(G) 1/2            -----(?)
(I) 1/4                
       -----(I) 1/2            -----(?)
                   -----(I) 2/3
                               -----(?)

I think the answer to a) is 1/2 x 2/3 x 3/4 but I am clueness about the rest. Please help! Thanks, Laura

 

[tkhunny]

You managed to confuse yourself. Your tree isn't growing fast enough.

Crime - Two branches.
Judge #1 - Two branches (Total of four at this level)
Judge #2 - Two branches (Total of eight at this level) You have only 4
Judge #3 - Two branches (Total of sixteen at this level)