Probability Problems
Math 160
February 4, 2008
1. Following example 4.7, write the entire sample space and find the probabilities of
getting exactly 0 heads, exactly 1 head, etc. What are the equally likely outcomes in this
problem?
2. You roll two dice, one red and one green.
a) Are “red 1” and “sum at least 10” independent events? Are they disjoint?
b) Are “red 6” and “sum at least 10” independent events? Disjoint?
c) There is one number k such that “red k” and “sum at least 10” are independent.
What is it? How can you see that it is the only one?
3. In a class of 24 students, 8 are wearing hats, and 10 are Biology majors. Three of the
Biology majors are wearing hats. If we pick one student at random (out of a hat), are
“random student wears a hat” and “random student is a Bio major” independent events?
Are they disjoint?
4. In a class of 24 students, 8 are wearing hats, and 6 had Cocoa Puffs for breakfast.
Having Cocoa Puffs and wearing a hat are independent. How many students are wearing
a hat and had Cocoa Puffs?
From http://www.ualberta.ca/MATH/gauss/fcm/BscIdeas/probability/DeMere.htm , with
some spelling corrected.
Historically recorded is the fate of a swinging Flemish renaissance gentleman, the
Chevalier de Mere. Around 1650, he suffered severe financial losses for assessing
incorrectly his chances of winning in certain games of dice. Contrary to the ordinary
gambler, he pursued the cause of his error with the help of Blaise Pascal. He reached
fame because in the process the area of probability was created. - Let us take a look at
what happened.
Among other things, the Chevalier systematically tried his luck with the following
two games.


First game: Roll a single die 4 times and bet on getting a six.
He assessed his chances of winning as follows. The chance of getting a 6 in a
single throw is 1 out of 6. Therefore, the chance of getting a 6 in 4 rolls is 4 times
1 out of 6; i.e. 2 out of 3.
Second game: Roll two dice 24 times and bet on getting a double six.
He assessed his chances of winning as follows. The chance of getting a double six
in one roll is 1 out of 36. Therefore, the chances of getting a double six in 24 rolls
is 24 times 1 out of 36; i.e. 2 out of 3.
To his painful surprise the Chevalier ended up losing badly with the second gamble.
He was desperate for an explanation, and so he sought help from one of the great
thinkers of his time, Blaise Pascal (1623-1662). After a careful analysis, Pascal was
able to spot the Chevalier's error.
5, 6. Find the actual probabilities of winning each of these games.
7. A friend claims that he can control whether a coin lands heads or tails. You are of
course skeptical, and ask him to prove it. On the first toss, you call heads; the coin lands
tails. On the second toss, you call heads; it lands tails.
a) What is the probability of your friend “winning” twice in a row? Would you
believe his claim based on what has happened so far?
b) What is the probability of your friend winning three tosses in a row? If you lost
again, would you then believe his claim?
c) How many times in a row would you have to lose in order to be pretty sure that
this friend can actually control the outcome of a coin toss? Explain your
reasoning.
8. You buy a box of a dozen chocolates, which claims to contain six chocolate-covered
cherries, and six chocolate-covered slugs. Not the good slugs, either. Picking three
candies at random, you find that all three contain slugs.
a) If the contents given on the box are actually correct, what would be the
probability of randomly choosing three slugs in a row? (Hint: unlike the last
problem, these events are not independent… BUT, you can still multiply, as long
as you multiply the right things.)
b) Would that give you reason to doubt the claim?
c) What if the fourth chocolate you tasted was also slug? Explain.