# Thread: Help with how much to allocate to varying probabilities to get specific probability?

1. ## Help with how much to allocate to varying probabilities to get specific probability?

Problem:
I must spend $1000. I have 3 strategies that have different Probabilities Of Success (POS). Strategy A POS = 70% Strategy B POS = 50% Strategy C POS = 25% How much do I spend on each strategy (must spend total of$1000) to have an overall POS of 65%?

4. ## Re: Help with how much to allocate to varying probabilities to get specific probabili

If X is success, then the probability of success $\displaystyle P(X)=0.65 = 0.70 \cdot P(A) + 0.50 \cdot P(B) + 0.25 \cdot P(C)$

If you let $\displaystyle b=0$ then you end up with $\displaystyle 0.7a + 0.50 \cdot 0 + 0.25 \cdot (1000-a)=650$

Law of total probability - Wikipedia, the free encyclopedia

5. ## Re: Help with how much to allocate to varying probabilities to get specific probabili

Originally Posted by dmbeas12
.65 = .70x + .5x + .25x
This is where you went wrong. Your solution assumes that you bet the same amount on each betting strategy. The percentages you've posted can be interpreted as conditional probabilities. Given a betting strategy A, B or C, what is the probability X that it success. This is where the law of total probability comes into the picture. If you have a sample space where the probability of all events add up to 1, then you can write the probability for an arbitrary event A as:

$\displaystyle P(A)=P(A|H_1)P(H_1) + P(A|H_2)P(H_2)+ ... + P(A|H_N)P(H_N)$, $\displaystyle P(H_1)+P(H_2)+...+P(H_N)=1$
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