11.8 Expected Value

11.8 - Expected Value

If the outcomes of an experiment have values E₁, E₂, E₃, E₄, . . . , E_n,

Then the Expected Value of the experiment is

E₁●P(E₂) + E₃●P(E₄) + E₅●P(E₆) + . . . . +E_n●P(E_n)

In other words . . . .
Expected Value = Sum of all the products of the outcomes multiplied by their respective probabilities.

Calculate Expected Gross Winnings of a Lottery Ticket

Example Calculate the expected gross winnings for the $1 BIG BEAR ticket with probabilities given below.

Prize Probability

$1 prize with probability 1/10

$2 prize with probability 1/10.64

$3 prize with probability 1/20

$10 prize with probability 1/166.67

$20 prize with probability 1/500

$30 prize with probability 1/750

$500 prize with probability 1/60,000

$5,000 prize with probability 1/240,000

The expected gross winnings are obtained by adding all of the products of the prize multiplied by the probability. This is calculated as

$1 × 1/10 + $2 × 1/10.64 + $3 × 1/20 + $10 × 1/166.67 +

$20 × 1/500 + $30 × 1/750 + $500 × 1/60,000 + $5,000 × 1/240,000

This may be input into a scientific calculator as

1 ÷ 10 + 2 ÷ 10.64 + 3 ÷ 20 + 10 ÷ 166.67 + 20 ÷ 500 + 30 ÷ 750 + 500 ÷ 60,000 + 5,000 ÷ 240,000 =

The expected gross winnings are $0.607 which rounds to $0.61 .

Thus, for each $1 ticket you purchase, you can expect to win 61 cents back on the average. You can, however, expect to lose a net 39 cents on each ticket.

Expected Winnings of a Lottery Ticket

The expected gross winnings of a lottery ticket is equal to the average gross amount that a person will win for each ticket purchased.

Example If the expected gross winnings of a $1 lottery ticket are $0.56, how much money will a person win expect to win by purchasing 1000 lottery tickets?

Since the person wins $0.56 per ticket, the person could expect to win 1000 x $0.56 = $560 by purchasing 1000 tickets. The expected net winnings are not $560 however since the 1000 tickets cost $1000.

By factoring in the purchase price, the net expected loss is $560 - $1000 = ($440).

Note that expected gross winnings do not indicate the actual amount of money but the expected amount of money that a person will win. The person could purchase the “big-winner” ticket and win much more than $560. Also, the person could win less than $560. Expected gross winnings merely indicates the “average” gross amount a person will win.

Lottery ticket “facts” are given here.

LOTTERY TICKET FACTS

The expected gross winnings of a lottery ticket indicates the average amount of winnings per ticket.

When a very large number of lottery tickets are purchased, the average gross winnings per ticket will be very close to the expected gross winnings of the ticket.

If all of the available tickets are purchased, the average gross winnings per ticket are equal to the expected gross winnings.

In the long run, a person experiences a net loss through the purchase of lottery tickets.

In the short run, a person “can sometimes beat the odds” and experience a net gain through the purchase of lottery tickets.

Assignment 8

Go to http://www.lottery.state.mn.us/wildhare.html. Use the lottery scratch game calculator shown below to or use a scientific calculator to calculate the expected gross winnings of a single one dollar ticket. Also, show the calculations required in addition to the answer, even if you used the scratch game calculator.

If you purchased 1000 of these tickets, what would your net loss be?

If there were a game with expected gross winnings of $0.74 for each dollar ticket sold and there were a total of one million dollar tickets distributed, what would it cost to buy all the tickets and how much would you win back?

Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the wildhare game above and split your earnings? Why or why not? In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?

How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

This Scratch-type Game Expected Winnings Calculator calculates the average amount you could expect to win per ticket. Fill in all the entries. If you don't have 15 prizes, then place ZERO for Amount of Prize and any NONZERO numbers for the odds for EACH non-prize. NO dollar signs! Scroll down and click on CALCULATE. What does all this mean?

Enter ZEROS for non-prizes	Enter NON-Zero values for non-prizes
Amount of Prize 1	Odds of winning prize 1 out of
Amount of Prize 2	Odds of winning prize 2 out of
Amount of Prize 3	Odds of winning prize 3 out of
Amount of Prize 4	Odds of winning prize 4 out of
Amount of Prize 5	Odds of winning prize 5 out of
Amount of Prize 6	Odds of winning prize 6 out of
Amount of Prize 7	Odds of winning prize 7 out of
Amount of Prize 8	Odds of winning prize 8 out of
Amount of Prize 9	Odds of winning prize 9 out of
Amount of Prize 10	Odds of winning prize 10 out of
Amount of Prize 11	Odds of winning prize 11 out of
Amount of Prize 12	Odds of winning prize 12 out of
Amount of Prize 13	Odds of winning prize 13 out of
Amount of Prize 14	Odds of winning prize 14 out of
Amount of Prize 15	Odds of winning prize 15 out of
Expected Winnings per Ticket	Calculated

Gambling Calculator Copyright 2003 Mike Sakowski - for educational use only
Most, if not all, scratch-type games use "odds" to stand for probabilities.
This calculator uses those "odds" as the probabilities input into the expected value formula.

What does all this mean? If you pay $1 for a ticket, and your expected winnings are $0.72, then you would expect a net loss of $0.28 per ticket. Hence, if you bought 100 tickets, you would experience a net loss of $28! The more you buy, the more you lose!