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 use the normal approximation to the binomial distribution, we can apply the Central Limit Theorem, which states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution.

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For a fair coin, the probability of getting a head ((p)) is 0.5, and the number of trials ((n)) is 400.

The mean ((\mu)) of the binomial distribution is given by (\mu = np), and the standard deviation ((\sigma)) is given by (\sigma = \sqrt{np(1-p)}).

a) For more than 180 heads:
[ P(X > 180) \approx P\left(Z > \frac{180 – np}{\sqrt{np(1-p)}}\right) ]
Substitute (n = 400), (p = 0.5), and calculate.

b) For less than 195 heads:
[ P(X < 195) \approx P\left(Z < \frac{195 – np}{\sqrt{np(1-p)}}\right) ]
Substitute (n = 400), (p = 0.5), and calculate.

Use the standard normal distribution table or a calculator with a standard normal distribution function to find the probabilities associated with the z-scores in both cases.