6.29 Suppose that a six-sided die is “loaded” so that any particular even-numbered face is twice as likely to land face up as any particular odd-numbered face.

Consider the chance experiment that consists of rolling this die.

a. What are the probabilities of the six simple events? (Hint: Denote these events by O1, … , O6.

Then P(O1) 5 p, P(O2) 5 2p, P(O3) 5 p, … , P(O6) 5 2p. Now use a condition on the sum of these probabilities to determine p.)

b. What is the probability that the number showing is an odd number? at most three?

c. Now suppose that the die is loaded so that the probability of any particular simple event is proportional to the number showing on the corresponding upturned face; that is, P(O1) 5

c, P(O2) 5 2c, … , P(O6) 5 6c. What are the probabilities of the six simple events? Calculate the probabilities of Part

(b) for this die.