Use the normal distribution to approximate the desired probability.
-Find the probability that in 200 tosses of a fair die, we will obtain at exactly 30 fives.
A) 0.0429
B) 0.0871
C) 0.1871
D) 0.0619

Question

Quizplus · Accepted Answer

We can use the normal approximation to the binomial distribution, since n = 200 is large enough and p = 1/6 is not too small or too large. Let X be the number of fives in 200 tosses, so X has a binomial distribution with n = 200 and p = 1/6. We need to find P(X = 30), which is approximately P(29.5 < X < 30.5) by the continuity correction.

The mean and standard deviation of X are given by μ = np = 200(1/6) = 33.33 and σ = sqrt(np(1-p)) = sqrt(200(1/6)(5/6)) = 4.21. Thus, we can standardize X as Z = (X-μ)/σ, which has a standard normal distribution.

Now, P(29.5 < X < 30.5) can be approximated by P((29.5-μ)/σ < Z < (30.5-μ)/σ) = P(-0.93 < Z < -0.67) using the standard normal table or calculator. This probability is approximately 0.0619, which is closest to choice D.

Use the Normal Distribution to Approximate the Desired Probability