Final answer:
To find the probability of receiving a certain amount of rainfall, use the z-score formula. For at most 41 inches of rain, the probability is approximately 43.32%. For at least 49 inches of rain, the probability is approximately 87.83%. For rainfall between 40 and 52 inches, the probability is approximately 58.18%.
Step-by-step explanation:
To find the probability of receiving a certain amount of rainfall, we need to convert the rainfall amounts to standard z-scores. The z-score formula is:
z = (x - μ) / σ
where x is the amount of rainfall, μ is the mean, and σ is the standard deviation. Once we have the z-scores, we can use a standard normal distribution table or a calculator to find the corresponding probabilities.
a) Probability of at most 41 inches of rain:
First, we calculate the z-score:
z = (41 - 42) / 6 = -0.1667
Using the standard normal distribution table, we find that the probability of z-score -0.1667 or less is approximately 0.4332. Therefore, the probability of receiving at most 41 inches of rain is 0.4332 or 43.32%.
b) Probability of at least 49 inches of rain:
First, we calculate the z-score:
z = (49 - 42) / 6 = 1.1667
Using the standard normal distribution table, we find that the probability of z-score 1.1667 or more is approximately 0.8783. Therefore, the probability of receiving at least 49 inches of rain is 0.8783 or 87.83%.
c) Probability of rainfall between 40 and 52 inches:
First, we calculate the z-scores:
z1 = (40 - 42) / 6 = -0.3333
z2 = (52 - 42) / 6 = 1.6667
Using the standard normal distribution table, we find that the probability of z-score -0.3333 or less is approximately 0.3707 and the probability of z-score 1.6667 or less is approximately 0.9525. Therefore, the probability of rainfall between 40 and 52 inches is the difference between these two probabilities: 0.9525 - 0.3707 = 0.5818 or 58.18%.