I had this problem on an exam a few weeks ago:
The persons $A$, $B$ and $C$ are playing a game. What the game is about is of no concern, the only thing we need to know is that the person who obtains the highest points wins and the following knowledge is available:
- $A$'s points is a possion distributed random variable with parameter 2.
- $B$'s points is a possion distributed random variable with parameter 3.
- $C$'s points is a possion distributed random variable with parameter 4.
- The points of each player are independent of eachother.
The outcome of the points were 1,2 and 3, but we do not know who got what points. What is the conditional probability, given this information, that it was person $C$ who won?
Let's for simplicity denote the random variables by $A,B$ and $C$. Then we know that $A\sim \text{poi}(2)$, $B\sim \text{poi}(3)$ and $C\sim \text{poi}(4)$.For $C$ to win we are looking for $$P(C = 3|A+B=3),$$ and then just use the formula for conditional probability/Bayes formula. Here is where I get stuck. This doesn't seem correct but I can't pinpoint exactly where its wrong.
Is there a sleak and easy-to-understand-way of solving this problem? I have a solution from my prof but I dont understand any of it. I can edit in that solution upon request.
Your $P(C=3\mid A+B=3)$ is not correct. It should be $P(C=3\mid\{A,B,C\}=\{1,2,3\})$.
We find: $$P(C=3\mid\{A,B,C\}=\{1,2,3\})P\left(\{A,B,C\}=\{1,2,3\}\right)=$$$$P(C=3\wedge\{A,B,C\}=\{1,2,3\})=$$$$P\left(A=1\wedge B=2\wedge C=3\right)+P\left(A=2\wedge B=1\wedge C=3\right)=$$$$\left[P(A=1)P(B=2)+P(A=2)P(B=1)\right]P(C=3)$$
Now find $P(\{A,B,C\}=\{1,2,3\})$ (a sum of $6$ probabilities) and apply:$$P(C=3\mid\{A,B,C\}=\{1,2,3\})=\frac{\left[P(A=1)P(B=2)+P(A=2)P(B=1)\right]P(C=3)}{P(\{A,B,C\}=\{1,2,3\})}$$