# Video: Finding the Probability of the Intersection of Independent Events in a Real Context

A system consists of two independently working components. The probabilities of failure of each component are 0.1 and 0.1. Assuming that the system fails if no component is functioning, find the probability that the system fails to function.

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### Video Transcript

A system consists of two independently working components. The probabilities of failure of each component are 0.1 and 0.1. Assuming that the system fails if no component is functioning, find the probability that the system fails to function.

The key here is that the working components are independent of one another. Let’s call the first component component A. When it comes to component A, one of two things will happen. It can fail or it cannot fail. When it comes to component B, the same two options apply. But we have to apply them if A fails or if A does not fail. We’ve been told that the system only fails if both components fail, which is the case that A fails and then B fails. The failure rate for component A is 0.1. And the failure rate for component B is 0.1.

Since a system failure happens if both A and B fail, the probability of this happening can be found by multiplying these two probabilities together. 0.1 times 0.1 equals 0.01. It was fine to draw out a probability tree here because this was a relatively simple problem. However, we do have a rule for working with the probability of independent events. For independent events, the probability of 𝐴 and 𝐵 occurring will be equal to the probability of 𝐴 times the probability of 𝐵. In our case, the probability that both components fail would be the probability that one fails times the probability that the other fails. And again that’s 0.01.