Probability Distributions and Expected Value

How can the Las Vegas casinos and other riverboat casinos make money on their table games, video poker games, and slot machines?

What about those "scratch off" games sponsored by the state and offered at convenience stores. How are they profitable?

7-11-21

Cost: $2
Features: WIN UP TO $10,000!
Add all 3 numbers for each game. If total is 7, 11, 21 in a single game, win prize shown for that game.

Probability Distributions and Expected Value
The answer to these questions is found in the study of probability distributions and expected value. Consider the following special holiday raffles:

Illinois Lottery's Millionaire Raffle

2008 St. Patrick's Day Millionaire Raffle

A limited number of 500,000 tickets will be sold.
Cost is $20 per-ticket.
Tickets will be sold in sequential order as sold throughout the state.
Your Best Shot at a Million!
Since only 500,000 tickets will be sold, your odds of winning a million dollars have never been better!
- 4 Prizes of $1,000,000, probability
  is 1 in 125,000!
- 5 prizes of $100,000 each,
  probability is 1 in 100,000!
- 500 prizes of $1,000 each,
  probability is 1 in 1000!
THAT’S 509 TOTAL AVAILABLE PRIZES WITH OVERALL PROBABILITIES ARE 1 IN 982!

What is the expected value of this raffle? On an average, how much money is each person purchasing a ticket losing. That can also be interpreted as how much money is Illinois make per ticket sold? Consider the following probability distribution table for this raffle.

Prize (Event)	x	Frequency	P(x)	*x P(x)**
$1,000,000	$999,980	4	4 / 500,000	$7.99984
$100,000	$99,980	5	5 / 500,000	$0.9998
$1,000	$980	500	500 / 500,000	$0.98
$0	-$20	499,491	499,491 / 500,000	- $19.97964
Totals		500,000	500,000 / 500,000	- $10.00

The expected value for the "Millionaire Raffle" is -$10.00. Every person purchasing a ticket loses $10 on the average. Conversely, Illinois makes $10 for every ticket sold. With 500,000 tickets sold, that means the ticket buyers are losing

The five million dollars loss by the ticket buyers is Illinois profit!

Another way to view this raffle is from a business perspective. Profit = Revenue - Cost.
Revenue = 500,000 tickets sold $20 each = 5000,000($20) = $10,000,000.
Costs = 4 winners of $1,000,000 = 4($1,000,000) = $4,000,000
               5 winners of $100,000 = 5($100,000) = $500,000
               500 winners of $1,000 = 500($1,000) = $500,000.
The total costs are $4,000,000 + $500,000 + $500,000 = $5,000,000
Profit = $10,000,000 - $5,000,000 = $5,000,000.
Note: The same answer as above.

Click on the hand to go to the General Education Mathematics raffle project.

Click on the hand to go to the Statistics raffle project.