When circuits boards used in manufacture of compact disc(CD) players are tested , the long-run percentage of defective is $5\text{%}$. Assume that there are $25$ boards in a sample. Find the probability that the number of defective boards is:
$a).$ At least 5:
I think since $20\leq n$ and $\theta \leq0.05$ it is good to use poisson approximation to binomial distribution
so $\lambda=25\times0.05=1.25 $, $P(X=x)=\dfrac{e^{-1.25}(1.25)^x}{x!}$
we have to find
$$P(x\geq 5 )=1-P(x<5)$$
But I saw in many website as well as Statistics - Probability of getting a number using combinations used only binomial distribution so I got problem why it is not used poisson distribution?
Can anyone help me is there anything wrong here?
The Poisson approximation only gives an approximate answer. To get an exact answer you must use a $B(25,0.05)$ distribution. The probability can be easily found by $$1-\sum_{k=0}^4\binom{25}{k}(0.05)^k(0.95)^{25-k}$$