[Math] Probability that the system works

Question

I have a system. That system has 4 components, and it can only works if 2 of the 4 components works correctly.

If every single component, independently,works correctly with a probability of 0,9. What is the probability that the system will work?

Answer 1

Hint: The probability that it works correctly is equal to $1$ minus the probability that it doesn't work correctly. The only cases where it doesn't work correctly are (I) exactly 0 of the 4 components work correctly, and (II) exactly 1 of the 4 components work correctly. So compute $P(\text{Case I})$ and $P(\text{Case II})$; the answer is $$ 1 - P(\text{Case I}) - P(\text{Case II}) $$

To deal with Case II, note that there are four ways that exactly one of them can work - the first one can work, or the second one can work, or the third, or the fourth.