Learning Targets
Calculator Goodies
TutorialsExpected value is what we expect to happen on average in the long run.
Example: Joe is playing a game which costs $5 to play. If he rolls a sum of 9 on the two dice, then he wins $20. Otherwise, he wins $0. Is this a fair game?
No. On the average, you would expect Joe to win only 4 out of every 36 times. That means he would win $80 out of 36 plays (4 times he'd win $20). But that would cost him 36*($5) = $180 to play 36 games. He'd definitely lose money in the long run.
Example: Joe is wanting to play a scratch ticket game with the following prize probability distribution.
How much would be a fair price for this game?
In the long run, Joe would expect to come out even for a fair price.
Since 16 is the common denominator for the probabilities, let's see what Joe would expect for 16 games.In 16 games, we'd expect on average:
$1 won 1/2 the time: ($1)*8 = $8
$5 won 1/4 the time: ($5)*4 = $20
$10 won 1/8 the time: ($10)*2 = $20
$50 won 1/16 the time: ($50)*1 = $50
Total winnings in 16 games: $98. That's $98/16 = $6.125 per game.A fair price would be $6.125.
Note: this is quickly found by ($1)(1/2) + ($5)(1/4) + ($10)(1/8) + ($50)(1/16) = $6.125
In general, we can find expected value as
where Σ stands for the sum, x is the value of an event, and P(x) is the probability of the event.
Example: Joe is thinking of playing the Washington State Lotto. Here is the current prize listing for this week.
(per $1 play)Is $1 a fair price?
EV = ($2,200,000)(1/6,991,908) + ($1,000)(1/27,100) + ($30)(1/516) + ($3)(1/28) = 0.5168
So Joe would expect to win only $0.52 on average for each time he plays.
No, $1 per play is not a fair price.Also, consider if Joe played 100 million times. He'd expect to win
$2,200,000 only (100,000,000/6,991,908) = 14.3 times. That's $31,464,944.
$1,000 only (100,000,000/27,100) = 3690 times. That's $3,690,036.
$30 only (100,000,000/516) = 193798 times. That's $5,813,953.
$3 only (100,000,000/28) = 3,571,428 times. That's $10,714,285.
In total, he'd win $51,683,218. Yet he'd have spent $100,000,000.
So he'd only win $0.52 per try on average and pay $1 per try. That's not a fair game.
