Chapter 3: Q. 3.30 (page 111)
Show that for any events and ,
Hint: Compute by conditioning on whether F occurs.
The result is that is weighted average between
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Let S = {1, 2, . . . , n} and suppose that A and B are, independently, equally likely to be any of the 2n subsets (including the null set and S itself) of S.
(a) Show that
P{A B} =
Hint: Let N(B) denote the number of elements in B. Use
P{A B} =P{A (B|N(B) = i}P{N(B) = i}
Show that P{AB = Ø} =
A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die, which is less than that appearing on the red die. That is, with B, Y, and R denoting, respectively, the number appearing on the blue, yellow, and red die, we are interested in .
(a) What is the probability that no two of the dice land on the same number?
(b) Given that no two of the dice land on the same number, what is the conditional probability that ?
(c) What is ?
An urn has r red and w white balls that are randomly removed one at a time. Let be the event that the ith ball removed is red. Find
a).
b).
c).
What is the probability that at least one of a pair of fair dice lands on , given that the sum of the dice is ?
In any given year, a male automobile policyholder will make a claim with probability pm and a female policyholder will make a claim with probability pf, where pf pm. The fraction of the policyholders that are male is α, 0 <α< 1. A policyholder is randomly chosen. If Ai denotes the event that this policyholder will make a claim in year i, show that P(A2|A1) > P(A1)
Give an intuitive explanation of why the preceding inequality is true.
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