Cumulative probability continued

[Brad_U]

cumulative probability continued...

Ok. I have 25 gumball machines with red or green gumballs. Each machine has a different percentage of each. Number 1 has 70% red, 30% green. Number 2 has 95% red, 5% green, and so on. I buy 1 gumball from each an drop it into a bag. Then reach into the bag and pull a random gumball. How to calculate probability Red or Green from the bag?

 

[CompuChip]

I would start calculating the expectation value of the green and red balls in the bag.

For example, if you draw one from each machine, you have on average (after repeating this experiment very very very often), (0,70 + 0,95 + ...) green balls in the bag, and (0,30 + 0,05 + ...) red ones.

Let's assume for this example that there are only two machines, so you will have 1,65 green ball and 0,35 red balls in a total of two.
If you draw one of the balls randomly, therefore, the probability of picking the green one is 1,65 / 2 = 0,825 and for the red one of course 0,175. (If you go back and check the calculation, this is basically the average of the probabilities, because we drew equal numbers of balls from each machine).

 

[Brad_U]

Thanks for the reply.