Answer to Question #86864 in Statistics and Probability for Cherrie Mae Magdalaga

Question

If only 55 percent kids can secure A grade in a paper, find the probability of at most 22 out of 1010 kids getting A grade in that paper.

Answer 1

Let X denote number of kids getting A grade among n=1010 kids, where only 55% kids can secure A grade in a paper.

X has a binomial distribution

$$Binomial (n=1010, p=0.55)$$ $$P(X=k)=\dbinom{n}{k} *p^k*(1-p)^{n-k}$$

The normal distribution can be used as an approximation to the binomial distribution, under certain circumstances, namely:

$X\sim Bin (n, p)$ and if n is large and/or p is close to 1/2, then X is approximately N(np, npq), where q=1-p.

$$p=0.55, q=1-p=1-0.55=0.45, n=1010$$ $$np=1010*0.55=555.5, npq=1010*0.55*0.45=249.975$$

$$z={{x-\mu} \over \sqrt{\sigma^2}}$$

$$x=22, z={{22-555.5} \over \sqrt{249.975}}=-33.7432$$

$$P(X\le22)=P(z\le(-33.7432)=0$$