Question

An electronic system contains three cooling components that operate independently. The probability of each component's failure is 0.05. The system will overheat if and only if at least two components fail. Calculate the probability that the system will overheat.

Answer 1

`Pr= 0.00725`

Explanation:

Given

`p = 0.05` ---- probability that each component fails

`n = 3`

Required

`P(System\ Overheats)`

We understand that the system will overheat if at least 2 component fails; Assume the components are: x, y and z

The events that the system will overheat are: xyz', xy'z, x'yz and xyz

Where ' means that the component did not fail, and the probability is 1 - p (i.e. complement rule)

So, we have:

`xyz' \to 0.05 * 0.05 * (1 - 0.05) = 0.002375`

`xy'z \to 0.05 * (1 - 0.05)* 0.05 = 0.002375`

`x'yz \to (1 - 0.05)* 0.05 * 0.05 = 0.002375`

`xyz \to 0.05 * 0.05 * 0.05 =0.000125`

So, the required probability is:

`Pr= 0.002375 +0.002375 +0.002375 + 0.000125`

`Pr= 0.00725`