If one stage has m options and the next has n, together they have m × n outcomes.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Permutations
A permutation counts arrangements, where order matters. Three books arrange in 3! = 6 ways.
Combinations
A combination counts selections, where order does not matter. It is a permutation with the orderings divided out.
How to choose
Ask whether swapping two chosen items changes the outcome. A president and vice-president, yes; a committee of two, no.
Expected value
Multiply each outcome by its probability and add. It is the long-run average result.
Fair games
A game is fair when the expected gain is zero, once the cost of playing is subtracted.
The multiplication principle
If one stage has m options and the next has n, there are mn outcomes. Nearly all counting reduces to this applied repeatedly, and a small tree confirms it.
Order matters, or it does not
Permutations count arrangements, combinations count selections. Ask whether swapping two chosen items changes the outcome; if not, divide out the orderings and you have a combination.
Expected value judges a decision
Multiply each outcome by its probability and sum. It is the long-run average, and comparing expected values is how two uncertain options are compared on the same footing.
Expected value ignores risk
A bet with positive expected value that could ruin you is still a poor decision, and insurance with negative expected value can still be rational. Expected value is one input to a decision, not the decision.
Step 2: Try It Yourself
Tap and try it out.
Arrangements
Selections
Arrangements has the most. It has 10 more than Selections.
Step 3: Watch an Example
One step at a time.
Watch Kofi Judge a Game
A game costs $2 and pays $10 with probability 0.1, nothing otherwise.
- Step 1
The expected payout is 10 × 0.1 + 0 × 0.9 = $1.
Step 4: Your Turn
Practice makes it stick.
The Committee
Problem 1 of 2
How many ways to choose 3 people from 6, when order does not matter?
The Game
Problem 2 of 2
Pays $10 with probability 0.1, costs $2. What is the expected gain, in dollars?
dollars
Count and Decide
1 of 8
What is 4 factorial?
2 of 8
What is 5C2?
3 of 8
What is 5P2?
4 of 8
3 shirts and 4 trousers. How many outfits?
5 of 8
Pays $20 with probability 0.25. Expected payout in dollars?
6 of 8
Expected payout $5, cost $5. Expected gain in dollars?
7 of 8
Sort each situation by whether order matters.
Tap something to move it.
- Empty
- Empty
8 of 8
What is 6C2?
Step 5: Quick Check
Show what you know.
Question 1 of 2
How many ways can 4 books be arranged in a row?
Question 2 of 2
What makes a game fair?
What You Learned
- Multiply the options at each stage to count outcomes.
- Permutations count arrangements; combinations count selections.
- A fair game has an expected gain of zero.