Use the appropriate normal distribution to approximate the resulting binomial distribution. A coin weighted so that the probability of obtaining a head in a single toss is 0.2 .If the coin is tossed 20 times, what is the probability of obtaining more than 10 heads? (Round your intermediate calculations to 2 decimal places.)
A) 0.9786
B) 0.0004
C) 0.0104
D) 0.0214
E) 0.9996

Question

Quizplus · Accepted Answer

To approximate the probability of obtaining more than 10 heads when tossing a weighted coin (with a probability of 0.2 for heads) 20 times using the normal distribution, we first calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = np and the standard deviation by σ = sqrt(np(1-p)), where n is the number of trials (20) and p is the probability of success (0.2). Thus, μ = 20*0.2 = 4 and σ = sqrt(20*0.2*0.8) = sqrt(3.2) ≈ 1.79. To find the probability of getting more than 10 heads, we convert this to a Z-score using the formula Z = (X - μ) / σ, where X is the target number of successes (10). So, Z = (10 - 4) / 1.79 ≈ 3.35. Looking up this Z-score in the standard normal distribution table or using a calculator gives a very small probability, indicating that getting more than 10 heads is highly unlikely. Hence, the closest match to this very small probability is option B) 0.0004.

Use the Appropriate Normal Distribution to Approximate the Resulting Binomial