I have this task to do:
A machine is composed of $3$ components, which function independently of
each other with probabilities $p_1$, $p_2$ and $p_3$, respectively. Calculate the probability that the machine works when
- the machine only works when all the components are working,
- the machine works when at least one of the components works.
Since I don't know what are the probabilities $p_1$, $p_2$ and $p_3$, I assume I would have to work on them, and not real numbers.
For the first case, I think it should be $p_1 \cdot p_2 \cdot p_3$.
When it comes to the second one, I'm confused. Should it be $1 – p_1 \cdot p_2$? It makes sense to me, but why wouldn't it be $1 – p_2 \cdot p_3$? Maybe I should include all of them? Any help would be greatly appreciated. Please, tell me if my first assumption is correct as well!
Best Answer
Your answer to the first part of the question is correct $$ P(\text{the machine works})=p_1p_2p_3. $$ The answer to the second part of the question can be obtained in the following way $$ P(\text{the machine works})=1-P(\text{the machine does not work}). $$ If the machine does not work, it means that all of the components failed. The probabability that a component fails is equal to $1-p_i$ for $i=1,2,3$. Using the independence, $$ P(\text{the machine does not work})=(1-p_1)(1-p_2)(1-p_3). $$ Hence, $$ P(\text{the machine works})=1-(1-p_1)(1-p_2)(1-p_3). $$