How Does the Probability of Failure Change in Synchronous Machine Operations?

[topgun08]

Question:
Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute
one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with
probability p2, all failure events being independent. Let the random variables X1, X2 denote the number of
runs until the first failure of M1, M2 respectively; thus X1, X2 have geometric distributions with parameters
p1, p2 respectively.
Let X denote the number of runs until the first failure of either machine. Show that X also has a geometric
distribution, with parameter p1 + p2 − p1p2

Attempt at an answer:
X1 has a geometric distribution of (1-p1)^i-1 * p1
X2 has a geometric distribution of (1-p2)^i-1 * p2

I'm confused an don't know how to proceed. Any help is appreciated.

 

[HallsofIvy]

Since this problem doesn't have anything to do with "Number Theory", I have it from that category.

 

[Ray Vickson]

Question:
Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute
one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with
probability p2, all failure events being independent. Let the random variables X1, X2 denote the number of
runs until the first failure of M1, M2 respectively; thus X1, X2 have geometric distributions with parameters
p1, p2 respectively.
Let X denote the number of runs until the first failure of either machine. Show that X also has a geometric
distribution, with parameter p1 + p2 − p1p2

Attempt at an answer:
X1 has a geometric distribution of (1-p1)^i-1 * p1
X2 has a geometric distribution of (1-p2)^i-1 * p2

I'm confused an don't know how to proceed. Any help is appreciated.

Look at the probability that neither M1 nor M2 has failed by n runs, for n = 1, 2, 3, ... .

RGV