p(first machine break down) = .1
p(second machine break down) = .05
p(third machine break down) = .025
p(first machine will not break down = 1 - .1 = .9
p(second machine will not break down = 1 - .05 = .95
p(third machine will not break down = 1 - .025 = .975
the probability that no machines will break down = .9 * .95 * .975 = .833625
the probability that all 3 will break down = .1 * .05 * .025 = .000125
the probability that the first machine will break down DISABLED_event_only= .1 * .95 * .975 = .092625
the probability that the second machine will break down DISABLED_event_only= .9 * .05 * .975 = .043875
the probabillity that the third machine will break down DISABLED_event_only= .9 * .95 * .025 = .021375
the probability that the first and second machine break down = .1 * .05 * .975 = .004875
the probability that the first and third machine break down = .1 * .95 * .025 = .002375
the probability that the second and third machine break down = .9 * .05 * .025 = .001125
total probability is 1 as it should be.
you had to do each scenario separately because the probability for each occurrence was different.
if they were the same, you could have used combination formulas as follows:
assume probability for each of the machines to break down was the same.
formula then would have been as follows:
p(0) = .9^3 * 1 = .729
p(1) = .1^1 * .9^2 * 3 = .243
p(2) = .1^2 * .9^1 * 3 = .027
p(3) = .1^3 * 1 = .001
total probability = 1 as it should be.
multiplication factors were based on combination formulas as follows:
number of possible ways to get 0 occurrences = c(3,0) = 3! / 0!*3! = 1
number of possible ways to get 1 occurrence = c(3,1) = 3! / 1!*2! = 3
number of possible ways to get 2 occurrences = c(3,2) = 3! / 2!*1! = 3
number of possible ways to get 3 occurrences = c(3,3) = 3! / 3!*0! = 1
your problem was more complex because each of the machines had a different probability of breaking down which is why each of the machines had to be treated differently.
you had 8 scenarios in each problem.
with your more complex problem you had to identify each of the scenarios uniquely because the probabilities for each were unique.
with my more simple problem you were able to combine 3 scenarios together for 1 occurrence and 2 occurrences because the probabilities were the same for all 3 machines.
this is how the different scenarios work out for both your complex problem and my simpler problem.
breakdowns machine 1 machine 2 machine 3
0 n n n
1 y n n
1 n y n
1 n n y
2 y y n
2 y n y
2 n y y
3 y y y