Cogito
Probability and Statistics · Chapter 3 · Lesson 3
Expected Value
What you would average over the long run.
12 problems · about 21 minutes · TEKS P.S.2.C, S-MD.B.5
What this lesson teaches
The student computes expected value and uses it to evaluate whether a game is fair.
- Expected value multiplies each outcome by its probability and adds.
- It is a long-run average and may never occur as a single result.
- A fair game has expected gain zero, once the cost of playing is subtracted.
Warm Up
Straightforward practice. Get the method working first.
5 problemsPays $40 with probability 0.1, else $0. Expected payout in dollars?
Answer 4
Why $4.
What makes a game fair?
Answer The expected gain is zero.
Why Zero expected gain.
Pays $10 with probability 0.5, else $0. Expected payout in dollars?
Answer 5
Why 10 × 0.5.
Pays $100 with probability 0.01, else $0. Expected payout?
Answer 1
Why 100 × 0.01.
Expected payout $5, cost to play $5. Expected gain?
Answer 0
Why 5 − 5.
Build It Up
The same ideas with more to keep track of.
3 problemsIs that game fair? 1 for yes, 0 for no.
Answer 1
Why Zero expected gain.
Expected payout $3, cost $4. Expected gain in dollars?
Answer -1
Why 3 − 4.
Pays $6 with probability 0.5 and $2 with probability 0.5. Expected payout?
Answer 4
Why 3 + 1.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich statements about expected value are true?
Answer It is a long-run average; It may be a value that never actually occurs
Why A die has expected value 3.5, a face it does not have.
Pays $50 with probability 0.02, else $0. Expected payout?
Answer 1
Why 50 × 0.02.
The Payout: A game pays $20 with probability 0.25 and $0 otherwise. What is the expected payout, in dollars?
Answer 5 dollars
Why $5.
The Die: What is the expected value of one roll of a fair six-sided die?
Answer 3.5
Why 3.5.