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.
1
(a) To maintain the typical house edge of an expected loss of − 19 , what should the payout
be if you wager one dollar?
1
Solution. To compute the fair payout that yields an expected loss of − 19 , consider the
following: There are 38 equally likely outcomes in American roulette. The probability of
4
winning the four-number bet is 38 , and the probability of losing is 34
38
.
Let x be the payout upon winning. The expected value E for a $1 wager is:
4 34 1
E= (x) + (−1) = −
38 38 19
Solving for x:
4 34 1
x− =−
38 38 19
4x − 34 = −2 (Multiply both sides by 38)
4x = 32
x=8
Thus, the fair payout is 8 .
(b) The actual payout for this bet is 8:1. What is the expected gain or loss from this wager?
Solution. With an 8:1 payout, the net gain is 8 dollars when you win, and the loss is 1
dollar when you lose.
4 34
E= (8) + (−1)
38 38
32 34
= −
38 38
2 1
= − = − ≈ −0.0526
38 19
1
Therefore, the expected loss is − or approximately $0.0526.
19
1
, Problem 2
In a dice game, you bet $2 on a number from 1 to 6. Three fair dice are rolled. If your
number appears on k ∈ {1, 2, 3} dice, you win 2k dollars (and keep your original wager). If
your number appears on none of the dice, you lose the wager. What is your expected net
profit or loss per round?
Solution. To compute the expected net profit, we define a random variable X representing
the net gain or loss for each round.
We first compute the probabilities P (k) for k = 0, 1, 2, 3 matches, using the binomial distri-
bution: k 3−k
3 1 5
P (k) = , k = 0, 1, 2, 3.
k 6 6
Explicitly,
125 75 15 1
P (0) = , P (1) = , P (2) = , P (3) = .
216 216 216 216
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.
We now calculate the expected value E[X]:
3
X
E[X] = 2kP (k) − 2P (0)
k=1
75 15 1 125
=2 +4 +6 −2
216 216 216 216
150 + 60 + 6 − 250 34 17
= =− =−
216 216 108
Thus, E[X] ≈ −0.1574 dollars.
On average, the player loses about $0.16 per round.
Problem 3
You are presented with two envelopes. One contains an unknown amount of money, and the
other contains 1.5 times that amount. You pick one envelope at random and find $120 inside.
Assuming equal probabilities, you reason that the other envelope might contain either $80
or $180.
(a) Compute the expected value if you switch.
2
Problem 1
In a “four number bet” in roulette, you win if any of the numbers 00, 0, 1, or 2 comes up.
1
(a) To maintain the typical house edge of an expected loss of − 19 , what should the payout
be if you wager one dollar?
1
Solution. To compute the fair payout that yields an expected loss of − 19 , consider the
following: There are 38 equally likely outcomes in American roulette. The probability of
4
winning the four-number bet is 38 , and the probability of losing is 34
38
.
Let x be the payout upon winning. The expected value E for a $1 wager is:
4 34 1
E= (x) + (−1) = −
38 38 19
Solving for x:
4 34 1
x− =−
38 38 19
4x − 34 = −2 (Multiply both sides by 38)
4x = 32
x=8
Thus, the fair payout is 8 .
(b) The actual payout for this bet is 8:1. What is the expected gain or loss from this wager?
Solution. With an 8:1 payout, the net gain is 8 dollars when you win, and the loss is 1
dollar when you lose.
4 34
E= (8) + (−1)
38 38
32 34
= −
38 38
2 1
= − = − ≈ −0.0526
38 19
1
Therefore, the expected loss is − or approximately $0.0526.
19
1
, Problem 2
In a dice game, you bet $2 on a number from 1 to 6. Three fair dice are rolled. If your
number appears on k ∈ {1, 2, 3} dice, you win 2k dollars (and keep your original wager). If
your number appears on none of the dice, you lose the wager. What is your expected net
profit or loss per round?
Solution. To compute the expected net profit, we define a random variable X representing
the net gain or loss for each round.
We first compute the probabilities P (k) for k = 0, 1, 2, 3 matches, using the binomial distri-
bution: k 3−k
3 1 5
P (k) = , k = 0, 1, 2, 3.
k 6 6
Explicitly,
125 75 15 1
P (0) = , P (1) = , P (2) = , P (3) = .
216 216 216 216
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.
We now calculate the expected value E[X]:
3
X
E[X] = 2kP (k) − 2P (0)
k=1
75 15 1 125
=2 +4 +6 −2
216 216 216 216
150 + 60 + 6 − 250 34 17
= =− =−
216 216 108
Thus, E[X] ≈ −0.1574 dollars.
On average, the player loses about $0.16 per round.
Problem 3
You are presented with two envelopes. One contains an unknown amount of money, and the
other contains 1.5 times that amount. You pick one envelope at random and find $120 inside.
Assuming equal probabilities, you reason that the other envelope might contain either $80
or $180.
(a) Compute the expected value if you switch.
2
