Download Expected Value

College Algebra
Sixth Edition
James Stewart  Lothar Redlin

Saleem Watson
9
Probability and
Statistics
9.4
Expected Value
Expected Value
Suppose that a coin has probability 0.8 of
showing heads.
• If the coin is tossed many times, we would expect
to get heads about 80% of the time.
• Now, suppose that you get a payout of one dollar
for each head.
• If you play this game many times, you would
expect on average to gain $0.80 per game
Expected Value—Definition
A game gives payoff a1, a2, … , an with
probabilities p1, p2, … , pn.
The expected value (or expectation) E of this
game is
E = a1p1 + a2p2 + … + an pn
E.g. 1—Finding Expected Value
A die is rolled.
• You receive $1 for each point that shows.
• What is your expectation?
E.g. 1—Finding Expected Value
Each face of the die has probability 1/6 of
showing.
• So, you get $1 with probability 1/6, $2 with
probability 1/6, $3 with probability 1/6, and
so on.
• Thus, the expected value is
 1
 1
 1
 1
 1
 1
E  1   2    3    4    5    6  
6
6
6
6
6
6
21

 3.5
6
E.g. 1—Finding Expected Value
So if you play this game many times, you will
make, on average, $3.50 per game.
E.g. 2—Finding Expected Value
In Monte Carlo, the game of roulette is
played on a wheel with slots numbered
0, 1, 2, …, 36.
• The wheel is spun, and a ball dropped in the
wheel is equally likely to end up in any one of
the slots.
• To play the game, you bet $1 on any number
other than zero.
• For example, you may bet $1 on number 23.
E.g. 2—Finding Expected Value
If the ball stops in your slot, you get $36.
• The $1 you bet plus $35.
• Find the expected value of this game.
E.g. 2—Finding Expected Value
You gain $35 with probability 1/37, and you
lose $1 with probability 36/37.
• Thus,
1
36
E   35 
  1
37
37
 0.027
E.g. 2—Finding Expected Value
In other words, if you play this game many
times, you would expect to lose 2.7 cents on
every dollar you bet (on average).
• Consequently, the house expects to gain
2.7 cents on every dollar you bet.