The annual rainfall in Cleveland, Ohio is approximately a...

The annual rainfall in Cleveland, Ohio is approximately a normal random variable with mean $40.2$ inches and standard deviation $8.4$ inches. What is the probability that (a) next year's rainfall will exceed 44 inches; (b) the yearly rainfalls in exactly three of the next seven years will exceed 44 inches? Assume that if $A_{i}$ is the event that the rainfall exceeds 44 inches in year $i$ (from now), then the events $A_{i}, i \geq 1$, are independent.

The annual rainfall in Cleveland, Ohio is approximately a normal random variable with mean $40.2$ inches and standard deviation $8.4$ inches. What is the probability that
(a) next year's rainfall will exceed 44 inches;
(b) the yearly rainfalls in exactly three of the next seven years will exceed 44 inches?
Assume that if $A_{i}$ is the event that the rainfall exceeds 44 inches in year $i$ (from now), then the events $A_{i}, i \geq 1$, are independent.

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Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve where the mean, median, and mode are all equal. It is used to model real-valued random variables whose distributions are not known, and many natural phenomena are approximated by it due to the Central Limit Theorem.

Standardization (Z-score)

Standardization involves converting a normally distributed variable into a standard normal variable (Z) by subtracting the mean and dividing by the standard deviation. This process enables the use of standard normal tables to find probabilities for values of the variable regardless of the original scale.

Independence of Events

Independence in probability means the occurrence of one event does not affect the probability of another occurring. This assumption is crucial when dealing with sequences of events, such as annual measurements, to ensure that the analysis of one event does not bias the results of another.

Binomial Distribution

The binomial distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent experiments, each with the same probability of success. It is particularly useful when analyzing scenarios where there are exactly two outcomes (success or failure) in each trial.

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