18.440 Probability and Random Variables
MIT Department of Mathematics
February 28, 2007
Due: Wednesday, March 7, 2007
Problem Set 4
Problems 1-6
Chapter 4, #17, #22, #36, #57
Chapter 4, Theoretical Exercises #25, #35
Problem 7 Game Show - Let’s Make a Deal
The first day of class, a variant of the classical game show was played. A $1 prize was
randomly placed in one of 6 envelopes. The contestant (an 18.440 student) was allowed
to pick any 3 of the 6 envelopes, which they thought contained the prize. If the prize
is in any of the envelopes the student picks, she wins the prize. The host (instructor)
knows which envelope contains the prize, so decided to show the student that 2 of her 3
picked envelopes are empty (the student frowned but was still hopeful since there was 1
remaining picked envelope that might contain the prize).
To make things more interesting, the instructor offered the following 4 choices, one at a
time, to the student. Justify your answers.
1 (a) Keep your pick.
(b) Give up your pick, and in exchange you are allowed to pick 2 of the remaining 3
envelopes.
2 (a) Keep your pick, and you will be given 10 cents.
(b) Give up your pick, and in exchange you are allowed to pick 2 of the remaining 3
envelopes.
3 (a) Keep your pick.
(b) Give up your pick, and in exchange you are allowed to pick 2 of the remaining 3
envelopes, and you will be given 25 cents.
4 (a) Keep your pick.
(b) Give up your pick, and in exchange you are allowed to pick all 3 of the remaining
envelopes.
A) Find the probability of winning for each of the cases above.
B) Determine the expected winnings for each of the cases above.
C) Which strategy is the overall optimal choice?
D) Suppose that the instructor does not know which envelope contained the prize. Two
of the student’s chosen envelopes are opened and shown to be empty. Conditional on the
event that two of the three envelopes are empty, what are the solutions to A,B, and C
above.
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