Lets say I have a bunch of horses in a race, and I know that probabilities of them winning are:
- $P(A) = 0.4$
- $P(B) = 0.3$
- $P(C) = 0.2$
- $P(D) = 0.08$
- $P(E) = 0.02$
QUESTION 1: How would I go about calculating the probability that the top 3 horses turn out to be arranged in this order:
- Horse $B$
- Horse $A$
- Horse $C$
QUESTION 2: How would I go about calculating the probability that Horses $A$, $B$, and $C$ are all in the top $3$, irrespective of order?
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Best Answer
The problem is that although you have given us the probability that each horse, we can't deduce the probability of a particular ranking without making further assumptions.
Here is a reasonable set of assumptions: let $S$ denote a subset $S \subset \{A,\dots,E\}$. Let $i$ denote an element of $S$. I assume that
The probability that $i$ comes first in a race among the elements of $S$ is $$ P = \frac{P(i)}{\sum_{j \in S} P(j)} = \frac{P(i)}{1 - \sum_{j \notin S} P(j)}. $$ That is, it is equal to the probability that $i$ comes first in a race of all horses given that the horses not in the race do not come first.
The probability of a ranking in which $i$ comes first is equal to the above probability multiplied by the overall probability of the ranking of $S \setminus \{i\}$. That is, the presence of the top horse does not affect the relative ranking of the remaining horses.
With these assumptions, we find that the overall probability of the order $B,A,C$ (the answer to question 1) is given by $$ P = P(B) \cdot \frac{P(A)}{1 - P(B)} \cdot \frac{P(C)}{1 - P(A) - P(B)} \approx 11.4 \%. $$