6. The number of major earthquakes in a year is approximately normally distributed with a mean of 20.8 and a standard deviation of 4.5 . a) Find the probability that each year there will be less than 21 earthquakes.

Step 1 ：We are given that the number of major earthquakes in a year is approximately normally distributed with a mean of 20.8 and a standard deviation of 4.5.

Step 2 ：We are asked to find the probability that each year there will be less than 21 earthquakes.

Step 3 ：To solve this problem, we need to use the properties of the normal distribution. The probability that a normally distributed random variable X is less than a certain value x is given by the cumulative distribution function (CDF) of the normal distribution.

Step 4 ：In this case, we need to find the CDF at x = 21. The CDF of a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) at a point x is given by the formula: \( P(X < x) = 0.5 * (1 + erf((x - \mu) / (\sigma * \sqrt{2}))) \) where erf is the error function.

Step 5 ：Substituting the given values into the formula, we get: \( P(X < 21) = 0.5 * (1 + erf((21 - 20.8) / (4.5 * \sqrt{2}))) \)

Step 6 ：Calculating the above expression, we get the probability as approximately 0.5177249324571197.

Step 7 ：Final Answer: The probability that each year there will be less than 21 earthquakes is approximately \(\boxed{0.5177}\).