4.5 Expectation

Three Machines Failing

I have three machines that each fail with probability 1/3. Assuming the machines operate (mutually) independently, what is the expected number of times I can run the three machines until all three of them fail in the same go?

Exercise 1

Let \(M_1, M_2, M_3\) be the events that machines 1, 2, and 3 fail.
Since these events are mutually independent, \(\Pr[M_1\cap M_2\cap M_3]=\Pr[M_1]\Pr[M_2]\Pr[M_3]=\frac{1}{27}\).
Then the mean time to failure is \(\frac{1}{1/27}=27\).