Game Of Theory EXAM Paper

GET 1018 / GEK 1544 The Mathematics of Games
Tutorial 1
1. In the movie “ 21 ”, Professor Rosa asked the student Ben the following question.
“There are three doors, and behind one of them is a car, while behind the other two are
goats. If you choose the door with the car behind it, you win the car. Now, say you choose
Door 1. The host then opens either Door 2 or Door 3, behind which is a goat. (The host
knows what is behind each door, and never opens the door with the car behind it.) The
host now gives you the choice: do you want to stick with Door 1, or switch to the other
door. What should you do? Or does it matter ? ”
Without hesitation Ben answered this correctly, which convinced Professor Rosa that Ben
would be a good addition to their “card counting team”. Explain why Ben’s answer is
correct, that is, by switching the choice rather than sticking with the original one, the
probability increases to 2/3 .
2. Solve Chevalier de Mere’s problem on determining the probability of obtaining one
or more double sixes in 24 rolls of a pair of honest dice.
3. Consider the basic property on probability :
P( C ∪ D) = P(C) + P(D) when C ∩ D = ∅
( ↑ i.e., no common events).
Show that P(A ∪ B) = P(A) + P(B) − P(A ∩ B) when A ∩ B ̸= ∅ .
4. Compute the probability on getting at least 1 six in 3 rolls of an honest die. Then
expose the flaw in the following argument, which attempts the same problem in the
“ positive direction” :
The probability of a six in any one roll is 1
Since we have 3 rolls, P (at least 1 six ) = 1
6 +
6 +
6 =
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