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.


  • 1
    $\begingroup$ It can be slightly simplified to $p_1p_2p_3(x)$. $\endgroup$ Jul 7, 2020 at 21:51
  • 1
    $\begingroup$ Because the events are independent, yes you can assume what you wrote. $\endgroup$
    – Matti P.
    Jul 8, 2020 at 12:32
  • $\begingroup$ @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. $\endgroup$
    – NoChance
    Jul 9, 2020 at 2:55

1 Answer 1


Your probabilities can be both independent or conditional.


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 $$P(\text{get profit} \ x) = P(\text{build product}| \text{sell for x})P(\text{sell for x}) = P(\text{build product})P(\text{sell for} \ x) = P(\text{have energy} | \text{have technical skills})P(\text{have technical skills})P(\text{sell for} \ x) = P(\text{have energy})P(\text{have technical skills})P(\text{sell for} x) = p_1p_2p_3.$$


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.

  • $\begingroup$ 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 $\endgroup$
    – NoChance
    Jul 9, 2020 at 9:06

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