Prize, cost and gain
Gain includes the cost of playing: Gain = prize − cost to play.
Win $8 and pay $3: your gain is $5.
Win nothing and pay $3: your gain is −$3 (a loss).
Type 1: is it fair?
A player gains 3 dollars with probability 0.3 and loses 2 dollars with probability 0.7. Is the game fair?
Step by step
- Use negative values for losses.
- Interpret the expected gain.
Final answer
No. In the long run, the player loses an average of 0.50 dollars per game.
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Find a fair fee or gain
Find a gain that makes the game fair
Lose 8 points with probability 0.5; gain 4 points with probability 0.4; gain k points with probability 0.1.
For a fair game, :
, so .
(c) A fair entry fee
What should the player pay so the game is fair?
Step by step
- The four equally likely results pay $0, $4, $4 and $8. Add the prizes and divide by 4.
- On average you receive $4. Pay $4 to leave no average gain or loss.
Final answer
Fair entry fee: $4.
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Revenue and compensation
Money, costs and compensation
Use the same method: identify numerical outcomes and their probabilities, multiply, add, then interpret.
A late delivery costs a shop 12 dollars compensation with probability 0.15; otherwise the cost is zero.
.
The long-run average compensation is 1.80 dollars per delivery.
Revenue after compensation; comparing options
For each outcome, subtract compensation from revenue first. Then calculate expected net revenue.
To compare two options, calculate the expected value for each and compare in context.
Average compensation
A shop pays €12 if a delivery is late. The chance is 0.15. Otherwise it pays €0.
Step by step
- Multiply each cost by its chance.
Final answer
Average compensation is €1.80 per delivery.
Exam-style question
Try it, then check
A game pays €0 with probability 0.75 or €8 with probability 0.25. It costs €3 to play. Is it fair? Find a fair fee. [4]
Step by step
- Find the average prize.
- Subtract the fee.
Final answer
Not fair: lose €1 on average. A fair fee is €2.