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.
b) Find the probability that each year there will be between 18 and 23 earthquakes.
Answer
Final Answer: The probability that each year there will be less than 21 earthquakes is approximately \( \boxed{0.518} \). The probability that each year there will be between 18 and 23 earthquakes is approximately \( \boxed{0.421} \).
Step 1 ：The problem is asking for the probability of a normally distributed random variable falling within certain ranges. To solve this, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF gives the probability that a random variable is less than or equal to a certain value.
Step 2 ：For part a), we need to find the probability that the number of earthquakes is less than 21. We can do this by calculating the CDF at 21.
Step 3 ：For part b), we need to find the probability that the number of earthquakes is between 18 and 23. We can do this by calculating the difference between the CDF at 23 and the CDF at 18.
Step 4 ：The mean of the distribution is 20.8 and the standard deviation is 4.5.
Step 5 ：The probability that the number of earthquakes is less than 21 is approximately 0.518.
Step 6 ：The probability that the number of earthquakes is between 18 and 23 is approximately 0.421.
Step 7 ：Final Answer: The probability that each year there will be less than 21 earthquakes is approximately \( \boxed{0.518} \). The probability that each year there will be between 18 and 23 earthquakes is approximately \( \boxed{0.421} \).
