Answer to Question #328082 in Statistics and Probability for robot

Question

An experiment has a binary outcome. Either success or failure. The probability of success is always 0.15. Suppose that the experiment is repeated 30 times.

a. Find out the probability that there are exactly 10 successes. 

b. Calculate the same probability using a Normal Approximation. Is the approximation good? If so why? If not why not?

Answer 1

a. $P(X=10)=C_{30}^{10}0.15^{10}(1-0.15)^{30-10}=0.0067.$

b. $\mu=30*0.15=4.5,\; \sigma=\sqrt{30*0.15*(1-0.15)}=1.96.$

$P(9.5<X<10.5)=P(\frac{9.5-4.5}{1.96}<Z<\frac{10.5-4.5}{1.96})=P(2.55<Z<3.06)=0.0043.$

Since $np=30*0.15=4.5<5$ , approximation is not good.