Help me out here. First off, let me say, I know what the right answers are. At least I think I do. What I understand is how it logically works.
Let's use sports as the medium:
1) Team A is trailing 3 games to 2 in a best of 7 series. Team A needs to win both games 6 and 7 in order to advance to the next round.
Team A has a 1/2 chance of winning game 6 and also has a 1/2 chance of winning game 7. 1/2 x 1/2 = 1/4, or 25% chance of advancing.
2 Team B is trailing 3 games to 2 in a best of 7 series. Team B needs to win both games 6 and 7 in order to advance to the next round.
Team B has a 2/5 chance of winning game 6 and a 3/5 chance of winning game 7. 2/5 x 3/5 = 6/25, or 24%
So the net is, Team A has a 25% chance of advancing when team B only has a 24% chance of advancing. I don't understand this. (1/2 + 1/2) is = to (2/5 + 3/5). To me that means that both teams should have an equal chance to advance... but obviously they don't.
Someone help me out here. Feel free to call me idiot as well, as long as you help.
That doesn't compute to me. In the 2nd game then, Team A has a 50% chance of winning where Team B has a 60% chance of winning then. So that should equally make up for the discrepancy for each teams' game 6s.
That still doesn't work to me. Flip flop it then. Give Team B the 60% chance in game 6 and the 40% chance in game 7. Their odds will still equate to 24%, but now they'd have the more likely opportunity to advance when compared to Team A.
That sorta makes sense to me, but I still struggle with it.
I used sports as the vehicle here, but theoretically, you could run the same scenario where the 7th game didn't depend on the 6th outcome. Like if you were running a simultaneous simulation and needed two things to occur to be the same... like a slot machine.
Quote:
There is less than 50 percent chance team B will advance so harder to get to a game 7 and therefore slightly lower overall chance of winning series
That still doesn't work to me. Flip flop it then. Give Team B the 60% chance in game 6 and the 40% chance in game 7. Their odds will still equate to 24%, but now they'd have the more likely opportunity to advance when compared to Team A.
Right, but it's why adding the probabilities together is irrelevant.
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Game 7 only happens if they win game 6. You'd sdd if they're ere mutually exclusive.
That sorta makes sense to me, but I still struggle with it.
I used sports as the vehicle here, but theoretically, you could run the same scenario where the 7th game didn't depend on the 6th outcome. Like if you were running a simultaneous simulation and needed two things to occur to be the same... like a slot machine.
If they were mutually exclusive, you'd add them, so the math is different.
Means Team A has a 10% greater chance to win game 6. Cannot get to game 7 unless they win game 6.
Means Team A has a 10% greater chance to win game 6. Cannot get to game 7 unless they win game 6.
Section, then flip flop it.
Give team B the 60% chance in game 1. I shouldn't have assigned specific values to each game. That's my bad. What I meant was Team B has a 60% chance to win one of the games and a 40% chance to win the other.
Section, then flip flop it.
Give team B the 60% chance in game 1. I shouldn't have assigned specific values to each game. That's my bad. What I meant was Team B has a 60% chance to win one of the games and a 40% chance to win the other.
Lower odds in either game negatively impacts the overall odds.
Extreme example:
Team needs to win game 6 & 7 to advance. They have a 100% chance of winning game 6 and a 0% chance of winning game 7.
By the way you calculate the odds, their chances are 50% of advancing. But the reality is their chances are 0%.
Team needs to win game 6 & 7 to advance. They have a 100% chance of winning game 6 and a 0% chance of winning game 7.
By the way you calculate the odds, their chances are 50% of advancing. But the reality is their chances are 0%.
That would be 0/100 x 100/100
100 x 0 = 0, so the math would correctly calculate a 0% chance.
That would be 0/100 x 100/100
100 x 0 = 0, so the math would correctly calculate a 0% chance.
Exactly. But In your example above, you were using addition rather than multiplication.
To calculate multiple probabilities we multiply the denominators to find the total outcomes in the new set. Sometimes a decision tree helps you see it best.
Let's start with team A.
Game 1 outcome Game 2 outcome Combined Outcome
P(win)=1/2 P(win)=1/2 P(win,win)=1/4
Win Loss WL
Win WW
Loss Loss LL
Win LW
Total outcomes = 4 (2 sets of 2 outcomes) and total events where A wins = 1.
Team B
Game 1 outcome Game 2 outcome Combined Outcome
P(win)=2/5 P(win)=3/5 P(win,win)=1/4
1. Win Loss WL
2. Loss WL
3. Win WW
4. Win WW
5. Win WW
6. Win Loss WL
7. Loss WL
8. Win WW
9. Win WW
10. Win WW
11. Loss Win LW
12. Win LW
13. Win LW
14. Loss LL
15. Loss LL
16. Loss Win LW
17. Win LW
18. Win LW
19. Loss LL
20. Loss LL
21. Loss Win LW
22. Win LW
23. Win LW
24. Loss LL
25. Loss LL
There are 25 possible outcomes (five possible outcomes in the first game, with two of them representing wins and three representing losses, and five for each of those five possible outcomes in the second game, with three of those representing wins and 2 representing losses). Of those 25 outcomes only 6 are WW, so you have P(WW)=6/25, or 24%.
[quote] he demonstrated the fallacy in your 0/100 + 100/100 = 1/2 + 1/2 thought process?
Very true.
The arithmetic means of [1/2,1/2] and [3/5,2/5] are the same, but their geometric means are not.
http://en.wikipedia.org/wiki/Geometric_mean - ( New Window )
Say they have a 50% chance of winning each game.
They have a 50% of not winning times a 50% of not winning game 7. So a 25% of losing both. There a 75% of winning either game.
Contestant A will make two passes... both times, he will have rocks flung at him by bystanders... giving him a 50% chance of surviving each pass, or 25% of survivng both.
Contestant B will make two passes... one time he will have nerf balls shot at him (95% chance of survival), the other time it will be live ammo (5% chance of survival) - so a 4.75% chance of surviving both.
The fact that he gets essentially a free pass against the nerf balls isn't going to help him against the bullets.