Howyadoit wrote:
The way to answer this is with probability estimates:
Chance it is a tactical race = .8
Chance Rupp wins tactical = .15
Win = .12
Chance Rupp medals tactical = .25
Medal = .2
Chance fast race = .2
Chance Rupp wins fast = .00
Win = 0
Medal = .1
Medal = .02
Total = .12 + .2 + .02 = .34 or 34%
Does medal exclude gold? Then you are saying 40% change of medal in a tactical race. Otherwise you are saying a 15% chance of gold and 10% change of either silver or bronze. I think that's way too optimistic for medals in the first case and way too optimistic for bronze in the second place.
Let's assume you mean, by medals, that includes 1, 2, or 3. Then the math below is wrong since the 0.2 already includes the 0.12, and your total should be 22%.
Regardless, I like the breakdown of your approach. Let's clarify the math so others can debate the probabilities. I'm leaving in your probabilities below and assuming the probability of medaling is split evenly between silver and bronze (with the leftovers after taking away the gold probability):
Let G, S, and B be the probabilities of Gold, Silver, and Bronze.
Prob of tactical race, P(T)=0.80
Prob of fast race, P(F)=0.20
Given a tactical race (T):
- Prob of Rupp Gold in a tactical race, P(G|T)=0.15
- Prob of Rupp Silver in a tactical race, P(S|T)=0.05
- Prob of Rupp Bronze in a tactical race, P(B|T)=0.05
- Prob of Rupp Medalling in a tactical race, then, is P(G|T)+P(S|T)+P(B|T)=0.25
Given a fast race (F):
- Prob of Rupp Gold in a fast race, P(G|F)=0
- Prob of Rupp Silver in a fast race, P(S|F)=0.02
- Prob of Rupp Bronze in a fast race, P(B|F)=0.08
- Prob of Rupp Medalling in a fast race, then, is P(G|F)+P(S|F)+P(B|F)=0.1
Total probability of winning a gold its P(T)P(G|T)+P(F)P(G|F)=(0.8)(0.15)+(0.2)(0)=0.12, or 12%
Total probability of winning a medal of any color is: P(T)[P(G|T)+P(S|T)+P(B|T)]+P(F)[P(G|F)+P(S|F)+P(B|F)]=(0.8)(0.25)+(0.2)(0.1)=0.22, or 22%
I'd go with different probabilities above, going with P(T)=90%
- Prob of Rupp Gold in a tactical race, P(G|T)=0.03
- Prob of Rupp Silver in a tactical race, P(S|T)=0.05
- Prob of Rupp Bronze in a tactical race, P(B|T)=0.15
- Prob of Rupp Gold in a fast race, P(G|F)=0.01
- Prob of Rupp Silver in a fast race, P(S|F)=0.02
- Prob of Rupp Bronze in a fast race, P(B|F)=0.08
Then probability of gold is (0.9)(.03)+(0.1)(0.01)=0.028, or 2.8%
Probability of a medal of any color is (0.9)(0.23)+(0.1)(0.11)=0.218, or 21.8%