Calculate probability of independent events

Question

Suppose I want to write a software. I know that:

• The probability I can overcome the technical difficulties is $p_1$
• The probability I can do all the work before I get too tired to complete it is $p_2$
• The probability I can get to sell it with profit $x$ is $p_3$

Assuming all events are independent, what would be my profit? Better yet could function be derived that would describe the profit value based on $p_1, p_2, p_3$.

Could I assume that $$ProbabiltyOfProfit_x(p_1, p_2, p_3)=p_1p_2p_3$$

One problem with this formula is that if I could not build the product, this is a probability of value $(1-p1*p2)$, then selling the product will not be logically possible.

Product selling, with probability $p3$ requires a product to exist in the first place. Accordingly I am not sure that $p3$ is independent of $p1,p2$.

How could this function be better formulated if correct?

Note: This is not homework.

Thanks.

Answer 1

Your probabilities can be both independent or conditional.

Independent:

In this case, $p_1$ is the probability that you can overcome the technical difficulties, $p_2$ is the probability that you have the energy to complete the work (independent of whether you have the technical skills or not), and $p_3$is the probability (independent of whether you finish the work or not) that the profit is $x$.

Then you can first apply the definition of conditional probabilities (P(A and B) = P(A|B)P(B)), and then use that due to independence we get $P(A|B) = P(A)$ ... so we get $$P(\text{get profit} \ x) = P(\text{build product}| \text{sell for x})P(\text{sell for x}) = P(\text{build product})P(\text{sell for} \ x) = P(\text{have energy} | \text{have technical skills})P(\text{have technical skills})P(\text{sell for} \ x) = P(\text{have energy})P(\text{have technical skills})P(\text{sell for} x) = p_1p_2p_3.$$

Conditional:

Your probabilities can also be conditional. Let's assume that $p_1$ and $p_2$ are independent, but $p_3$ is conditional on the first two events already having occured. That is, $p_3$ is the probability of selling for profit $x$ if the software has been built. Then we get

$$P(\text{get profit} \ x) = P(\text{sell for x}| \text{product built})P(\text{product built}) = p_3p_2p_1,$$

so the answer is the same.