Calculation of a most probable value based on multiple probability distributions

Given are multiple probability distributions for a random variable.

Is there a way to calculate the most probable value based on these distributions?

Example:

Random Variable: Number of hairs on the head of a person
Probability Distribution 1: Number of hairs depending on gender
Probability Distribution 2: Number of hairs dependent on age
Probability Distribution 3: Number of hairs dependent on hair color

Now I want to calculate the most probalble value for the number of hairs for a 34 years old, blonde woman. Does this sound reasonable?

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-1 No. You should first study to understand the basics, then you'll know if it makes sense to seek for software to do the calculations. Here, it doest make much sense. For one thing, "expected value" is not the same as "most probable". – leonbloy May 8 '12 at 15:37
Ok, I reformulated the question. The example should demonstrate the problem well in my eyes. – Heinrich May 8 '12 at 17:05

1 Answer

It sounds like you are in the realm of conditional probability. In particular, let $N$ be the number of hairs, $G$ the gender, $A$ the age and $H$ the hair color. You have the following distributions:

$P(N|G)$, $P(N|A)$ and $P(N|H)$.

You are interested in $P(N|G, A, H)$. The issue is that you don't quite have enough data to make observations. What you really need is how gender age and hair color correlate with each other. For example, you can assume that age and gender are independent. As well, maybe gender and hair color are independent as well. Assuming you don't count gray hair, age and hair color can be considered independent too.

From the probability chain rule:

$P(N|G,A,H) = \frac{P(N,G,A,H)}{P(G|A,H)P(A|H)P(H)}$

Now the terms in the denominator by independence can reduce to $P(G)$, $P(A)$ and $P(H)$ each of which you can determine just by looking at statistics, say $P(G)$ is just $1/2$ for male female, etc. For the numerator you just need to take a large sample of people and write down the number of occurances of each quadripule $(N,G,A,H)$.

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