Alice, Bob, Carol, and Dave are being given some money, but they have to share it in a very particular way: 

• First, Alice will be given an integer number of dollars A, chosen uniformly at random between 1 and 100 (inclusive).

• Then Bob receives from Alice an integer number of dollars B, chosen uniformly at random between 1 and A (inclusive). 

• Then Carol receives from Bob an integer number of dollars C, chosen uniformly at random between 1 and B (inclusive). 

• Then Dave receives from Carol an integer number of dollars D, chosen uniformly at random between 1 and C (inclusive). 

a. Draw the Bayes net with variables A, B, C, D that corresponds to this process. 

b. Write down an exact formula for P(B = b), the probability that Bob receives b dollars. (Your answer may contain a summation.) 

c. Show that P(A = a | B = b) is proportional to 1/a. 

d. Suppose Dave receives $5 from Carol. You would like to estimate the probability P(A > 50 | D = 5). Your first attempt uses rejection prior sampling, generating the following samples from the prior: 

What is the estimated probability for the query based on these samples? 

e. Your next attempt uses likelihood weighting, generating the following samples:

Calculate the weights for each sample and find the estimated value for the query.

Transcribed Image Text:

(A=52, B=21, C = 10, D=5) (A=34, B=21, C=6, D=3) (A=96, B=48, C = 12, D=2) (A=13, B12, C = 10, D=1) (A=54, B12, C = 11, D=6) (A=91, B=32, C=31, D=29)