Question

Use the appropriate normal distribution to approximate the resulting binomial distribution.

A coin is weighted so that the probability of obtaining a head in a single toss is 0.7. If the coin is tossed 20 times, find the following probabilities. (Round your answers to four decimal places.)

(a) fewer than 10 heads


(b) between 10 and 15 heads, inclusive


(c) more than 19 heads

Answer 1

To approximate the binomial probability for a weighted coin tossed 20 times with a 0.7 chance for heads, we calculate the mean and standard deviation for the binomial distribution and apply the normal approximation with continuity correction, finding probabilities by using z-scores and a standard normal distribution table.

Step-by-step explanation:

To approximate the binomial distribution, we can use the normal distribution if certain conditions are met. For a coin weighted so that the probability of obtaining a head in a single toss is 0.7, if the coin is tossed 20 times, we can calculate the following probabilities.

Calculating the mean and standard deviation

First, we find the mean (μ) and standard deviation (σ) for the binomial distribution.

• Mean (μ) = np = 20 * 0.7 = 14
• Variance (σ2) = npq = 20 * 0.7 * 0.3 = 4.2
• Standard deviation (σ) = √(npq) = √(4.2) ≈ 2.0494

Using the normal approximation to the binomial distribution:

1. For fewer than 10 heads: We calculate P(X < 9.5) since we need to adjust by 0.5 for the continuity correction.
2. For between 10 and 15 heads, inclusive: We calculate P(9.5 ≤ X ≤ 15.5).
3. For more than 19 heads: We calculate P(X > 19.5).

Then, we use the z-score formula to find these probabilities using a standard normal distribution:

Z = (X - μ) / σ

For each of the scenarios above, we would find the respective z-scores and consult a standard normal distribution table or use software/calculators to find the probabilities to four decimal places.