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Math 402 Study Guide

Problem 1
In a “four number bet” in roulette, you win if any of the numbers 00, 0, 1, or 2 comes up
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To compute the fair payout that yields an expected loss of − 19
, consider the
following: There are 38 equally likely outcomes in American roulette
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winning the four-number bet is 38
38

Let x be the payout upon winning
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(b) The actual payout for this bet is 8:1
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With an 8:1 payout, the net gain is 8 dollars when you win, and the loss is 1
dollar when you lose
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0526
38
19
Therefore, the expected loss is −

1
or approximately $0
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19
1

Problem 2
In a dice game, you bet $2 on a number from 1 to 6
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If your
number appears on k ∈ {1, 2, 3} dice, you win 2k dollars (and keep your original wager)
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What is your expected net
profit or loss per round?
Solution
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We first compute the probabilities P (k) for k = 0, 1, 2, 3 matches, using the binomial distribution:
   k  3−k
5
3
1
, k = 0, 1, 2, 3
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216

A $2 stake yields a net profit of 2k dollars when k ≥ 1, and a loss of $2 when k = 0:
(
2k, k = 1, 2, 3,
X=
−2, k = 0
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1574 dollars
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16 per round
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One contains an unknown amount of money, and the
other contains 1
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You pick one envelope at random and find $120 inside
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(a) Compute the expected value if you switch
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We identify the two possibilities for the unknown envelope:
80 =

120
1
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5×120 (if 120 is the smaller amount)
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2
2
2
Because you already hold $120, this suggests an expected gain of $10 by switching
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Solution
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Let X denote the smaller of the two amounts
prepared
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However, the actual probability distribution over X determines which scenario is more likely,
and the equal probability assumption generally does not follow from a coherent prior
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This leads to a contradiction of the conservation of
expected value and thus reveals the logical flaw
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, n, and mean µ = E[X]
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(a) Show that the conditional distribution for X is:
pˆk =

kpk
µ

Solution

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