A fair coin is tossed 400 times. Using normal approximation to the binomial, find the probability that a head will occur a) More than 180 times b) Less than 195 times.

 To find the probability of certain outcomes in a binomial experiment using normal approximation, we can apply the Central Limit Theorem. The Central Limit Theorem states that as the sample size (number of trials) increases, the sampling distribution of the sample mean (or sum) approaches a normal distribution, even if the original distribution is not normal. In this case, we have a binomial experiment (coin tosses), and we will use normal approximation to calculate the probabilities of getting heads.

Given data:

• Coin tosses (n) = 400
• Probability of getting a head in a single toss (p) = 0.5 (since the coin is fair)

Step 1: Calculate the mean and standard deviation of the binomial distribution: For a binomial distribution, the mean (μ) and standard deviation (σ) are given by:

μ = n * p

σ = sqrt(n * p * (1 - p))

Substitute the values:

μ = 400 * 0.5 = 200

σ = sqrt(400 * 0.5 * (1 - 0.5)) = sqrt(200) ≈ 14.14

Step 2: Apply normal approximation to calculate probabilities:

a) Probability of getting more than 180 heads (x > 180): We need to find P(x > 180), where x is the number of heads obtained in 400 coin tosses.

To apply normal approximation, we convert the binomial distribution to a standard normal distribution using the z-score formula:

z = (x - μ) / σ

Where: x = 180 (number of heads we are interested in) μ = 200 (mean of the binomial distribution) σ = 14.14 (standard deviation of the binomial distribution)

z = (180 - 200) / 14.14 ≈ -1.414

Now, we find the probability P(x > 180) using the standard normal table or a calculator:

P(x > 180) = 1 - P(x ≤ 180)

Using the standard normal table or calculator, we find P(z ≤ -1.414) ≈ 0.0793.

Therefore, P(x > 180) = 1 - 0.0793 ≈ 0.9207 (or 92.07%).

b) Probability of getting less than 195 heads (x < 195): We need to find P(x < 195), where x is the number of heads obtained in 400 coin tosses.

To apply normal approximation, we convert the binomial distribution to a standard normal distribution using the z-score formula:

z = (x - μ) / σ

Where: x = 195 (number of heads we are interested in) μ = 200 (mean of the binomial distribution) σ = 14.14 (standard deviation of the binomial distribution)

z = (195 - 200) / 14.14 ≈ -0.3535

Now, we find the probability P(x < 195) using the standard normal table or a calculator:

P(x < 195) = P(z < -0.3535)

Using the standard normal table or calculator, we find P(z < -0.3535) ≈ 0.3616.

Therefore, P(x < 195) ≈ 0.3616 (or 36.16%).

In conclusion, using normal approximation to the binomial distribution, we calculated the probabilities of getting heads in the following scenarios:

a) The probability of getting more than 180 heads in 400 coin tosses is approximately 0.9207 (or 92.07%).

b) The probability of getting less than 195 heads in 400 coin tosses is approximately 0.3616 (or 36.16%).

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