A k-out-of-n system is one that will function if and only if at least $k$ out of the $n$ individual components in the system function. Assume that individual components function independently of each other. Assume also each individual component functions with probability $0.9$
Determine the long-run proportion of 3-out-of-5 systems that will function.
The binomial distribution looks to be a good fit here.
Summarising the data:
$p$ (the probability of success) $= 0.9$
$n$ (the total number of trials) $= 5$
$x$ (the number of successes needed) $= 3$
3-out-of-5 means the system needs at least $3$ components to function which means the system will function at $3$ components, $4$ components $5$ components.
Which in turn means the answer will be $1 - prob$($0$ components + $1$ components + $2$ components)
Therefore $1 - 0.0815$
Is this correct?