Introduction to Games of Chance
Primary 4 · Mathematics
Introduction to Games of Chance
Lesson 2 of 4
30 min
Let's begin
Ishimwe tosses three coins at the same time. Keza makes up a game: "If all three coins match, I win. If they are mixed, you win." Ishimwe agrees, but after a few rounds he is winning almost every time. Is Keza's game fair?
To find out, we must list every possible outcome.
A tree diagram helps you list every outcome without missing any. The first coin can be H or T. For each of those, the second coin can be H or T. For each of those, the third coin can be H or T.
There are 2 × 2 × 2 = 8 outcomes, and each one is equally likely.
| Number of heads | Outcomes | How many |
|---|---|---|
| 3 heads | HHH | 1 |
| 2 heads | HHT, HTH, THH | 3 |
| 1 head | HTT, THT, TTH | 3 |
| 0 heads | TTT | 1 |
| Total | 8 |
Keza wins when all three coins match: HHH or TTT. That is only 2 outcomes out of 8. Ishimwe wins with every mixed outcome: 6 out of 8. Ishimwe has three times as many chances, so the game is not fair.
Change the rules: Keza wins with 2 or 3 heads, and Ishimwe wins with 0 or 1 head. Now Keza has 3 + 1 = 4 winning outcomes and Ishimwe has 3 + 1 = 4. Equal chances, so the new game is fair.
Tip
Branch, branch, branch. Every coin splits each branch in two, so 1 coin gives 2 outcomes, 2 coins give 4 and 3 coins give 8. List them all before you decide if a game is fair.
Work each one in your exercise book first. Then tap Show answer to check.
How many outcomes are there when you toss two coins? List them.
Answer: 4: HH, HT, TH and TT.
With three coins, how many outcomes have exactly 1 head?
Answer: 3: HTT, THT and TTH.
With three coins, how many outcomes have at least 2 heads?
Answer: 4: HHH, HHT, HTH and THH.
With three coins, is getting 3 heads likely or unlikely?
Answer: Unlikely. Only 1 outcome out of 8 has 3 heads.
Uwase wins if exactly 2 coins show heads. Byiringiro wins if exactly 1 coin shows heads. They toss again if all coins match. Is this fair?
Answer: Yes. Each has 3 winning outcomes (2 heads: 3; 1 head: 3).
What you learned today
Sources and credits
The three-coin game and the way to make it fair were written by Ganzaa. The stories, Rwandan examples, diagrams and practice questions were written by Ganzaa.
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