Can I use a Poisson distribution for non-normal binomial data?

Hello all. I'm working on the following question:

Fifteen percent of the items produced by a machine are defective. Out of 15 items chosen at random,
a)what is the probability that exactly 3 items will be non-defective?

Now this is a binomial distribution but; pn = 85/100 (15) = 12.75 and qn = 15/100 (15) = 2.25 (right?)
With qn being so low this isn't suitable for normal approximation and so the binomial distribution can't work here right? (I tried calculating it and realized z scores seemed much too high)

So can I use the Poisson distribution for this problem? And if I do, do I have to calculate upper and lower limits (3.5 & 2.5 respectively) and then subtract one from the other like I would for the binomial distribution, or will I just use one X value of 3 for the equation?
I haven't been able to find much help online about using the Poisson distribution for just two outcomes.

Let me know if I'm missing something here.
Really appreciate any help or advice!


Ambassador to the humans
I think you're confused on the rules of thumb. The binomial can be used in any case where it's appropriate. The normal approximation is typically assumed to be 'good enough' if p*n > 5 and (1-p)*n > 5. That is what you were checking so the normal distribution approximation to the binomial distribution failed. There are other rules of thumb for when it's alright to use the poisson approximation for the binomial but is there a reason you *don't* want to use the binomial distribution directly?