I am solving a problem in probability related to Joint distribution. Let’s say I am playing shooting arrow at a target and after every shot, I choose a number from {2,5}. I shoot twice and everytime I hit the target, I choose 2 and if I do not I choose 5.Let’s suppose the probability of hitting the target is 1/2. So the set of all possible outcomes are {(2,2), (2,5), (5,2), (5,5)}. Let A be the first number chosen so A belongs to {2,5}? Let B be the product of the chosen numbers. Now I am supposed to draw a joint probability distribution table for A and B. And I am not sure if I am doing the correct thing.
My attempt:
So B = {4, 10, 25} and my table looks like this:
I am not sure if the corresponding joint probabilities are correct. For example, in order to calculate P(A =2, B =10), my reasoning was. There are four total elements in our sample space. The probability that the first number chosen is 2 AND the product of the two numbers chosen is 10 is only satisfied by (2,5). So, I thought the probability should be 1/4.
Am I doing anything wrong here?