"a system consists of n identical component each of which is operational with probability p independent of others and a system is operational if more than half of its component working correctly, find the number p in which a system with 5 component have better performance than a system with 3 components?"
how can i solve this problem? i get into this solution but i don't think its correct : $\frac{4}{5} + \frac{5}{5} > \frac{2}{3}$ it means that the probability of 4 components or 5 components of the first system working correctly must be more than 2 components of the second, that way the performance of the first is better, is it correct?
Hint: Letting $X\sim\text{Binomial}(5,p)$ and $Y\sim\text{Binomial}(3,p)$ you want to find $p$ such that $\mathsf P(X\geq 3) \gt \mathsf P(Y\geq 2)$ since for the system to operate with $5$ components, we need at least $3$ to work and for the system to operate with $3$ components, we need at least $2$ to work.