18.5. Linearity of Expectation 777
(a)What is the probability that Bruce breaks exactly 2 out of the 5 boards that are
placed before him?
(b)What is the probability that Bruce breaks at most 3 out of the 5 boards that are
placed before him?
(c)What is the expected number of boards Bruce will break?Problem 18.10.
A news article reporting on the departure of a school official from California to
Alabama dryly commented that this move would raise the average IQ in both states.
Explain.
Class Problems
Problem 18.11.
Here’s a dice game with maximum payoffk: make three independent rolls of a fair
die, and if you roll a six
no times, then you lose 1 dollar; exactly once, then you win 1 dollar; exactly twice, then you win 2 dollars; all three times, then you winkdollars.For what value ofkis this game fair?^6
Problem 18.12. (a)Suppose we flip a fair coin and letNTTbe the number of flips
until the first time two consecutive Tails appear. What is ExŒNTTç?
Hint:LetDbe the tree diagram for this process. Explain whyDcan be described
by the tree in Figure 18.8. Use theLaw of Total Expectation18.4.5.
(b)LetNTHbe the number of flips until a Tail immediately followed by a Head
comes up. What is ExŒNTHç?
(c)Suppose we now play a game: flip a fair coin until eitherTTorTHoccurs.
You win ifTTcomes up first, and lose ifTHcomes up first. SinceTTtakes 50%
(^6) This game is actually offered in casinos withkD 3 , where it is called Carnival Dice.