# Price Calculation based on probabilities

This is a strange question, it might be easy for all you math wizards or it even may be impossible. If you don't understand what I mean let me know so I can change the way I post the question.

Product A can be built using Product B.

To get Product B I have to transform product C or Product D.

I have a 50% chance to get product B by transforming product C and 25% to get product B by transforming product D.

Is there any way to predict the cost of production for Product A knowing the price for Product C and D

Edit 1 There are no transformation costs, only the costs for the materials

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Suppose you go with the strategy "buy C and attempt a transformation to B until you get B". Then with probability 1/2, this will cost C, with probability $1/2(1-1/2)$, cost $2C$, with probability $1/2^n$ cost $nC$. Thus the expected price will be

$C\sum_{n=1}^\infty \frac{n}{2^n}=2C$

Similarly, the strategy "buy D and attempt a transformation to B until you get B" will have expected price

$B_D+D\sum_{n=1}^\infty \frac{n3^{n-1}}{4^n}=4D$

It's no coincidence that the expected price is just the cost divided by the probability of success; this follows from the properties of the binomial distribution.

You'll want to compare these two and see which is minimal and run with that strategy.

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There is no cost of transformation, how would that change your calculation? Sorry for the newbiesh question. – Nuno Furtado Nov 3 '10 at 17:19
As per their definitions, $B_C$ and $B_D$ will be equal to zero... – j.c. Nov 4 '10 at 10:02

What i thinks is that if you go for Producing Product A from Product C it will cost you double the total price of Product C and if you Produce Product A from Product D it will cost you 4 times the total Price of Product D according to Probabilities respectively :)

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