Imagine you and a friend are playing a video game. You both put in $10 to win a $20 prize. The goal is to reach 10 wins, but the power goes out when you are winning 9 to 8.
Since someone has to win 10 points within the next two rounds, the possible outcomes are:
- You win, then You win (You win the prize)
- You win, then Friend wins (You win the prize) - we are understanding this in this article - because you are already winning 9, if you win 1 more you won the game.
- Friend wins, then You win (You win the prize)
- Friend wins, then Friend wins (Friend wins the prize)
This is the single hardest part of probability to wrap your head around, so don't sweat it. You are right: in the real world, there are only 3 paths.
The problem is that those 3 paths don't have the same weight.
If you just count the 3 paths as "1 win for me, 1 win for me, 1 win for friend," you might think your odds are 2 out of 3 (66%). But that’s a trap.
The "Weight" Problem
Think of it like a path in the woods that splits:
- Path 1 (Left): Leads straight to your house. (50% chance you take this)
- Path 2 (Right): Leads to another fork in the road. (50% chance you take this)
- Path 2a: Leads to your house. (Half of that 50% = 25%)
- Path 2b: Leads to your friend's house. (Half of that 50% = 25%)
Even though there are only 3 destinations, the first destination (Path 1) is "fatter" because it happens earlier.
Why Fermat "Invents" the 4th Possibility
To make the math easy, Fermat wants every path to be the same size (same weight).He says: "Let's pretend you have to walk past your house and keep going to a second fork, even though you’re already home."
By forcing the game to go 2 rounds, he turns that "Fat Path 1" into two equal skinny paths:
- Path 1 becomes: (You win, then You win) + (You win, then Friend wins).
- Both of these lead to you winning.
The Math Logic
- Realistic (3 paths): 50\100 + 25\100 + 25\100 = 100\100
- Fermat (4 paths): 25\100 + 25\100 + 25\100 + 25\100 = 100\100
He "creates" the 4th possibility (breaking your 50% win into two 25% wins) just so he can say, "Look, 3 out of 4 paths are yours!" It’s just a clever way to avoid doing fractions in your head.
In short: You don't have to use his way, but if you don't, you have to remember that your first "win" is twice as likely to happen as your friend's "win."
Does that "fat path" vs "skinny path" idea make it clearer why we can't just count them as 1:1?
