13.4 Expected Value
• Understand the meaning of expected
value.
• Use expected value to solve applied
problems.
• Calculate the expected value of lotteries
and games of chance.
• Assume that an experiment has
outcomes numbered 1 to n with
probabilities P1, P2, P3, …, Pn. Assume
that each outcome has a numerical
value associated with it and these are
labeled V1, V2, V3, …, Vn. The
expected value of the experiment is
(P1 • V1) + (P2 • V2) + (P3 • V3) + … +
(Pn • Vn).
– A game that has an expected value of O is
called a fair game. A game in which the
expected value is not 0 is an unfair game.
Roulette
• There are many ways to bet on the 38
numbers of a roulette wheel. One simple
betting scheme is to place a $1 bet on a
single number. If you win, the casino pays
you $35 (you also keep your $1 bet). If
not, you lose the $1. What is the
expected value of this bet?
Solution
• There are 38 likely numbers that can occur.
• Your number comes up and the value to you is
•
•
•
+$35. The probability of this is 1/38.
Your number doesn’t come up and the value to
you is -$1. The probability of this is 37/38.
The expected value of this bet is:
(1/38 · 35) + (37/38 · -1) = -0.0526
On average, the casino expects you to lose
slightly more that 5 cents for every dollar you
bet.
Classwork/Homework
• Classwork – Page 766 (5 - 9 odd, 21, 23)
• Homework – Page 766 (6 – 10 even, 22,
24)
Dice Table
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)