Question

For the binomial experiment of tossing a fair coin 250 times, use the normal distribution approximate the probability that: the coin lands a heads at least 110 times.

Answer 1

To approximate the probability that a fair coin lands heads at least 110 times when tossed 250 times, we can use the normal distribution. Using the mean and standard deviation of the binomial distribution, we calculate the z-score and find the probability using a standard normal distribution table or a calculator.

Step-by-step explanation:

To approximate the probability that the coin lands heads at least 110 times when tossing a fair coin 250 times, we can use the normal distribution. The mean of the binomial distribution is given by μ = np = 250 * 0.5 = 125, and the standard deviation is given by σ = √(npq) = √(250 * 0.5 * 0.5) = 8.84. To use the normal distribution, we need to calculate the z-score, which is given by z = (x - μ) / σ. In this case, we want to find the probability that the coin lands heads at least 110 times, so we need to calculate the z-score for x = 110. Once we have the z-score, we can use a standard normal distribution table or a calculator to find the probability.

Here's the step-by-step calculation:

1. Calculate the z-score:
   z = (110 - 125) / 8.84 = -1.69
2. Find the probability using a standard normal distribution table or a calculator:
   P(Z ≥ -1.69) = 1 - P(Z ≤ -1.69)
3. Look up the corresponding probability in the standard normal distribution table or use a calculator to get the final answer.

For example, using a standard normal distribution table, we find that P(Z ≤ -1.69) = 0.0446. Therefore, P(Z ≥ -1.69) = 1 - 0.0446 = 0.9554. So, the probability that the coin lands heads at least 110 times is approximately 0.9554.