17.8. Mutual Independence 721
(a)What is the probability that you will get shot if he pulls the trigger?(b)Suppose he pulls the trigger and you don’t get shot. What is the probability
that you will get shot if he pulls the trigger a second time?
(c)Suppose you noticed that he placed the two shells next to each other in the
cylinder. How does this change the answers to the previous two questions?
Problem 17.3.
State and prove a version of the Law of Total Probability that applies to disjoint
eventsE 1 ;:::;Enwhose union is the whole sample space.
Problem 17.4.
State and prove a version of Bayes Rule that applies to disjoint eventsE 1 ;:::;En
whose union is the whole sample space. You may assume then-event Law of Total
Probability, Problem 17.3.
Class Problems
Problem 17.5.
There are two decks of cards. One is complete, but the other is missing the Ace
of spades. Suppose you pick one of the two decks with equal probability and then
select a card from that deck uniformly at random. What is the probability that you
picked the complete deck, given that you selected the eight of hearts? Use the
four-step method and a tree diagram.
Problem 17.6.
Suppose you have three cards: A~, A, and a Jack. From these, you choose a
random hand (that is, each card is equally likely to be chosen) of two cards, and let
Kbe the number of Aces in your hand. You then randomly pick one of the cards
in the hand and reveal it.
(a)Describe a simple probability space (that is, outcomes and their probabilities)
for this scenario, and list the outcomes in each of the following events:
1.ŒK1ç, (that is, your hand has an Ace in it),- A~is in your hand,
- the revealed card is an A~,