Download Wheel of Fortune problems D. Penneys On the game show Wheel of

D. Penneys
Wheel of Fortune problems
On the game show Wheel of Fortune, players spin the following wheel:
(source: Wikipedia http://en.wikipedia.org/wiki/File:Wheel_of_Fortune_Round_1_
template_Season_31.png). The problems in this section all have to do with spinning the
wheel. Assume each spin is independent, and landing on each of the 24 large sectors is
equally likely, where one sector is evenly divided into 3 smaller sectors; the one million dollar
sector and the two Bankrupt sectors on either side of it are 1/3 the size of all the other
sectors. This means it is 1/3 as likely to land on one of these small sectors as it is to land
on a large sector.
Two sectors are labelled “1/2 CAR”. If you land on both, you win a car.
Problem 1. (1) On a given spin, what is the probability of landing on a red sector?
(2) On a given spin, what is the probability of landing on one million dollars? (Assume
that the one million dollar sector is 1/3 the size of all the other sectors, as are both the
Bankrupt sectors on either side.)
(3) On a given spin, what is the probability of landing on Bankrupt?
(4) On average, how many spins does it take to land on Bankrupt?
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Problem 2. You spin once, and let X be the random variable which counrs the number of
0’s (not to be confused with O’s). What is E[X]? What is Var(X)?
Problem 3. Given that you landed on a green sector, what is the probability you landed on
one million dollars?
Problem 4. You spin the wheel twice. Given that you landed on a $700 sector and an $800
sector, what is the probability you landed on a red and a yellow sector?
Problem 5. Suppose you spin the wheel three times.
(1) How many ways are there to land on 3 red sectors?
(2) Given that you landed on three $700 sectors, what is the probability you landed on three
distinct colors?
Problem 6. Suppose you spin the wheel twice, except if you land on Bankrupt the first time,
you don’t get another spin.
(1) What’s the probability you win a car?
(2) What is the probability you land on a red sector at least once?
Problem 7. Suppose you spin the wheel up to 3 times, except if you land on Bankrupt, you
get no more additional spins.
(1) What is the probability you spin exactly twice?
(2) What is the probability you land on a red sector at least once?
(3) What’s the probability you win a car?
Problem 8. You keep spinning until you win a car. How many spins do you expect this will
take?
Problem 9. You spin the wheel 5 times. You land on a red sector twice, a blue sector twice,
and a purple sector once. What is the probability of this event?
Problem 10. Suppose you spin the wheel twice and your sectors total 1500.
(1) What are the possible dollar amounts for both spins?
(2) What are the possible pairs of colors?
(3) What is the probability one of your spins lands on a blue sector?
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Hints
Problem 4: Note that you can land on red then yellow, or yellow then red! This also applies to
Problem 5.
Problem 8: Let X1 be the random variable which counts the number of spins until you get the first
“1/2 CAR”. Let X2 be the random variable which counts the number of spins after
you get the first “1/2 CAR” until you get the second “1/2 CAR”. Let X = X1 + X2 .
Now calculate E[X].
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