Chapter 20: The House Edge: Expected Values
On average, how much do you win when you gamble?
Thought Question
Which is more important to you - knowing how much it will cost to gamble over the long-run,
or knowing you could potentially win a lot of money?
Multistate lotteries have enormous jackpots but very small probabilities of winning. Roulette
has much larger probabilities of winning but smaller jackpots.
How can we determine which game to play?
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Expected Value
Terminology
• Expected value: average of the possible outcomes, weighted by the probability of each
outcome.
How to Calculate the Expected Value
Let a1 , a2 , . . . , ak be the possible outcomes and p1 , p2 , . . . , pk be the associated probabilities.
Then
expected value = a1 p1 + a2 p2 + . . . ak pk .
Example: Tri-State Lottery
The “Straight” wager from the Pick 3 game of the Tri-State Daily Numbers is as follows. You
pay $0.50 and choose a three-digit number. The state chooses a three-digit number at random
and pays you $250 if your number is chosen.
How many outcomes for lottery numbers are there, and what is the probability of each outcome?
1000 outcomes, each with probability 1/1000
Determine a probability model for your winnings.
Outcome
Probability
$0
0.999
$250
0.001
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If you place 1000 bets, roughly many times will you win $250? Roughly how many times will
you win $0?
You will win $250 1 time and win $0 999 times.
Calculate your expected winnings.
(0)(999) + (1)(250)
= $0.25.
1000
999
1
+ (0)
= $0.25.
expected winnings = (250)
1000
1000
average winnings over 1000 plays =
Including the cost of the bet, how much will you win/lose on average?
You will lose $0.25 on average.
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Law of Large Numbers
If a random phenomenon with numerical outcomes is repeated many times independently, the
mean of the observed outcomes approaches the expected value.
So for many independent repetitions:
• the proportion of each possible outcome will be close to its probability, and
• the average outcome obtained will be close to the expected value.
So the expected value calculates the long-run average.
On average, will you win money or lose money when you gamble?
You will lose money on average - that is why casinos stay in business!
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Example: Michigan Classic Lotto 47
How to play: select 6 numbers from 1 to 47
Cost: $1 per play
Winnings:
Match
3 of 6
4 of 6
5 of 6
6 of 6
Else
Prize
$5
$100
$2,500
Jackpot (starts at $1 million)
$0
Probability
0.019856
0.001146
0.000023
0.00000009313
0.978976
Calculate the expected winnings, assuming a Jackpot of $1 million.
expected winnings = (5)(0.019856)+(100)(0.001146)+(2, 500)(0.000023)+(1, 000, 000)(0.00000009313)+
(0)(0.978976) ≈ $0.365
Using your previous answer, calculate the expected winnings after including the cost of the game.
expected winnings ≈ $0.365 − 1 = −$0.635
What value does the Jackpot need to be to break even?
Solve (5)(0.019856)+(100)(0.001146)+(2, 500)(0.000023)+(x)(0.00000009313)+(0)(0.978976)−
1=0
x ≈ $7, 823, 687, so the jackpot needs to be about $7,823,687 to break even.
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