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.