I'm trying to solve this mathematical puzzle, but I can't seem to find a solution.
The problem states that we are a financial trader and we have a list of trades that we performed during the day. Each trade has associated a potential profit and the probability of success of that trade. Each trade has two possible outcomes: it either yields its profit with probability pi, or yields nothing with probability 1-pi.
Given the list of profits and probability, we are required to compute the probability that the trades we performed will bring us a profit greater than a number 1000.
The only constraints that are given are that the profits are integer numbers between 1 and 1000.
The example that is provided to better explain the puzzle is the following one:
probabilities = {0.5, 0.4}
profits = {700, 800}
The probability that we make a profit greater than 1000, in this case, is 0.2 since the only way is that both the trades yield their profits indeed 0.5 * 0.4 = 0.2. For this small example is easy to find a solution by hand but for a bigger problem, I was only able run simulations in order to find an approximate solution. I think that Discrete Random Variable theory is associated with this problem but my bad I do not know how to use it in order to solve it. I would like to know if exist an alternative approach that can bring an exact solution ?