Cogito
Integrated Math 2 · Chapter 8 · Lesson 3
Counting and Fair Decisions
Counting outcomes, and using probability to decide.
12 problems · about 21 minutes · S-CP.B.9, S-MD.B.5
What this lesson teaches
The student counts outcomes with permutations and combinations and evaluates fairness using expected value.
- Multiply the options at each stage to count outcomes.
- Permutations count arrangements; combinations count selections.
- A fair game has an expected gain of zero.
Warm Up
Straightforward practice. Get the method working first.
5 problemsHow many ways can 4 books be arranged in a row?
Answer 24
Why 24.
What makes a game fair?
Answer The expected gain is zero once the cost is subtracted.
Why Zero expected gain.
What is 4 factorial?
Answer 24
Why 4 × 3 × 2 × 1.
What is 5C2?
Answer 10
Why 120 ÷ (2 × 6).
What is 5P2?
Answer 20
Why 5 × 4.
Build It Up
The same ideas with more to keep track of.
3 problems3 shirts and 4 trousers. How many outfits?
Answer 12
Why 3 × 4.
Pays $20 with probability 0.25. Expected payout in dollars?
Answer 5
Why 20 × 0.25.
Expected payout $5, cost $5. Expected gain in dollars?
Answer 0
Why A fair game.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each situation by whether order matters.
Answer Order matters: Awarding first and second place, Setting a 4-digit passcode · Order does not matter: Choosing 3 players for a team, Picking 2 pizza toppings
Why Ask whether swapping two chosen items changes the result.
What is 6C2?
Answer 15
Why 720 ÷ (2 × 24).
The Committee: How many ways to choose 3 people from 6, when order does not matter?
Answer 20
Why 20.
The Game: Pays $10 with probability 0.1, costs $2. What is the expected gain, in dollars?
Answer -1 dollars
Why −$1.