Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-Estimate the probability that in 200 rolls of a balanced die, the number of sixes is between 36 and 45 inclusive.
A)0.3239
B)0.3305
C)0.2914
D)0.2639

Question

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-Estimate the probability that in 200 rolls of a balanced die, the number of sixes is between 36 and 45 inclusive. 
A)0.3239
B)0.3305
C)0.2914
D)0.2639

Quizplus · Accepted Answer

To estimate this probability using the normal distribution as an approximation to the binomial distribution, we first need to find the mean (μ) and standard deviation (σ) of the binomial distribution. For a binomial distribution, μ = np and σ = sqrt(np(1-p)), where n is the number of trials (in this case, 200 rolls), and p is the probability of success on each trial (getting a six, which has a probability of 1/6).μ = np = 200 * (1/6) = 33.33σ = sqrt(np(1-p)) = sqrt(200 * (1/6) * (5/6)) ≈ 7.07Next, we convert the binomial problem into a normal distribution problem. We are asked to find the probability of getting between 36 and 45 sixes, inclusive. To use the normal approximation, we apply continuity correction, adjusting our range to 35.5 (one half below 36) and 45.5 (one half above 45).Now, we convert these values to z-scores:z1 = (35.5 - 33.33) / 7.07 ≈ 0.31z2 = (45.5 - 33.33) / 7.07 ≈ 1.72Using standard normal distribution tables or a calculator, we find the probabilities corresponding to these z-scores and subtract the smaller from the larger to find the probability of being between these two values.P(0.31 < Z < 1.72) ≈ P(Z < 1.72) - P(Z < 0.31) ≈ 0.9573 - 0.6268 = 0.3305Therefore, the correct answer is B) 0.3305.

Estimate the Indicated Probability by Using the Normal Distribution as an Approximation