Chapter 16 Events and Probability Spaces558
Problems for Section 16.2
Exam Problems
Problem 16.1. (a)What’s the probability that 0 doesn’t appear amongkdigits
chosen independently and uniformly at random?
(b)A box contains 90 good and 10 defective screws. What’s the probability that
if we pick 10 screws from the box, none will be defective?
(c)First one digit is chosen uniformly at random fromf1;2;3;4;5gand is re-
moved from the set; then a second digit is chosen uniformly at random from the
remaining digits. What is the probability that an odd digit is picked the second
time?
(d)Suppose that yourandomlypermute the digits1;2;;n, that is, you select
a permutation uniformly at random. What is the probability the digitkends up in
theith position after the permutation?
(e)A fair coin is flippedntimes. What’s the probability that all the heads occur
at the end of the sequence? (If no heads occur, then “all the heads are at the end of
the sequence” is vacuously true.)
Problem 16.2.
Here’s a variation of Monty Hall’s game: the contestant still picks one of three
doors, with a prize randomly placed behind one door and goats behind the other
two. But now, instead of always opening a door to reveal a goat, Monty instructs
Carol torandomlyopen one of the two doors that the contestant hasn’t picked. This
means she may reveal a goat, or she may reveal the prize. If she reveals the prize,
then the entire game isrestarted, that is, the prize is again randomly placed behind
some door, the contestant again picks a door, and so on until Carol finally picks a
door with a goat behind it. Then the contestant can choose tostickwith his original
choice of door orswitchto the other unopened door. He wins if the prize is behind
the door he finally chooses.
To analyze this setup, we define two events:
GP:The event that the contestantguesses the door with theprize behind it on his
first guess.
OP:The event that the game is restarted at least once. Another way to describe
this is as the event that the door Carol firstopens has aprize behind it.