# Calculate probability of independent events

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.

• It can be slightly simplified to $p_1p_2p_3(x)$. Jul 7, 2020 at 21:51
• Because the events are independent, yes you can assume what you wrote. Jul 8, 2020 at 12:32
• @MattiP., thank you for your comment. I made an edit, to argue whether the events are truly independent or not...You may want to have a look. Jul 9, 2020 at 2:55

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.

• Welcome to Mathematics Stack Exchange. I think the sell event must be depndent on the production event. Accordingly your second part is the correct one. Thanks Jul 9, 2020 at 9:06