15.2. Expected Value and Payoffs http://www.ck12.org
Solution:Using the concept of weighted average, weight each of Owen’s grades by the weight of the category.
0. 78 · 0. 3 + 1 · 0. 25 + 0. 74 · 0. 20 + 0. 90 · 0. 20 + 1 · 0. 05 = 0. 862
Owen gets an 86.2%.
Example B
Courtney plays a game where she flips a coin. If the coin comes up heads she wins $2. If the coin comes up tails she
loses $3. What is Courtney’s expected payoff each game?
Solution:The probability of getting heads is 50% and the probability of getting tails is 50%. Using the concept of
weighted averages, you should weight winning 2 dollars and losing 3 dollars by 50% each. In this case there is no
initial cost to the game.
2 · 0. 50 − 3 · 0. 50 =− 0. 50
This means that while sometimes she might win and sometimes she might lose, on average she is expected to lose
about 50 cents per game.
Example C
Paul is deciding whether or not to pay the parking meter when he is going to the movies. He knows that a parking
ticket costs $30 and he estimates that there is a 40% chance that the traffic police spot his car and write him a ticket.
If he chooses to pay the meter it will cost 4 dollars and he will have a 0% chance of getting a ticket.
Is it cheaper to pay the meter or risk the fine?
Solution:Since there are two possible scenarios, calculate the expected cost in each case.
Paying the meter: $4·100%=$4
Risking the f ine: $0·60%+$30·40%=$12
Risking the fine has an expected cost three times that of paying the meter.
Concept Problem Revisited
In a game that pays you triple your bet if you roll a six and double your bet if you roll a five, the expected return on
a one dollar wager is:
$0·^23 +$2·^16 +$3·^16 =^56
If you spend $1 to play the game and you play the game multiple times, you can expect a return of^56 of one dollar
or about 83 cents on average.
Vocabulary
Aweighted averageis an average that multiplies each component by a factor representing its frequency or probabil-
ity.
Theexpected valueis the return or cost you can expect on average, given many trials.
Thepayoffof a game is the expected value of the game minus the cost.
Guided Practice
- What is the payoff of a slot machine that costs a dollar to play and pays out $5 with probability 4%, $10 with
probability of 2%, and $30 with probability 0.5%? - What is the expected value of an experiment with the following outcomes and corresponding probabilities?