Problem 16 - A First Course in Probability

Question

The annual rainfall (in inches) in a certain region is normally distributed with \(\mu=40\) and \(\sigma=4 .\) What is the probability that starting with this year, it will take more than 10 years before a year occurs having a rainfall of more than 50 inches? What assumptions are you making?

Step-by-Step Solution

Answer

The probability that it takes more than 10 years before a year occurs having a rainfall of more than 50 inches, given the annual rainfall is normally distributed with \(\mu = 40\) and \(\sigma = 4\), is approximately 93.86%. We assume that the annual rainfall follows a normal distribution and that the rainfall in each year is independent of the rainfall in previous years.

Step 1: Recall the Z-score formula

The Z-score, also known as the standard score, is a measure that describes the number of standard deviations an element is from the mean of a distribution. In this case, we are given that the annual rainfall follows a normal distribution with a mean \(\mu = 40\) and a standard deviation \(\sigma = 4\). We can calculate the Z-score of a value x using the following formula: \(Z = \frac{x - \mu}{\sigma}\)

Step 2: Find the Z-score corresponding to 50 inches of rainfall

We want to find the probability that it takes more than 10 years to have a year with more than 50 inches of rainfall. To do this, we need to find the Z-score corresponding to 50 inches of rainfall. We can use the formula from Step 1: \(Z = \frac{50 - 40}{4} = 2.5\)

Step 3: Find the probability of having a year with more than 50 inches of rainfall

Now, we need to find the probability that the annual rainfall is more than 50 inches in a given year. Since the Z-score we calculated in Step 2 is 2.5, we can look that up in a Z-table or use standard normal cumulative distribution function to find the probability. Using the Z-table or the standard normal cumulative distribution function, we find that \(P(Z>2.5) \approx 0.0062\). This means there is a 0.62% chance that the annual rainfall will be more than 50 inches in a given year.

Step 4: Find the probability of not having a year with more than 50 inches of rainfall

Now, we want to find the probability that it takes more than 10 years for a year with more than 50 inches of rainfall to occur. First, we find the probability of not having a year with more than 50 inches of rainfall in a given year. Since there is a 0.62% chance that the annual rainfall will be more than 50 inches, the probability of not having a year with more than 50 inches of rainfall is: \(1 - 0.0062 = 0.9938\)

Step 5: Find the probability of not having a year with more than 50 inches of rainfall for 10 consecutive years

We now have the probability of not having a year with more than 50 inches of rainfall in a given year (0.9938). To find the probability of this happening for 10 consecutive years, we simply multiply the probabilities together: \(0.9938^{10} \approx 0.9386\) So, there is approximately a 93.86% chance that it takes more than 10 years for a year with more than 50 inches of rainfall to occur. As for the assumptions being made in this problem, we assume that the annual rainfall follows a normal distribution and that the rainfall in each year is independent of the rainfall in previous years.

Key Concepts

Normal distribution

The normal distribution, often called the Gaussian distribution, is a fundamental concept in the field of probability and statistics. It is known for its bell-shaped curve and is characterized by its mean (\(\mu\)) and standard deviation (\(\sigma\)). In a normal distribution, most data points cluster around the mean, resulting in the curve's peak at the center.
Key features of the normal distribution include:

• Symmetry around the mean
• A mean that represents the location of the center of the distribution
• A standard deviation that indicates the spread or width of the distribution
• The area under the curve represents probabilities

The area under the entire curve adds up to 1, encapsulating all potential outcomes. Understanding the normal distribution is vital because many real-world phenomena tend to naturally form a normal curve when measured.

Z-score

The Z-score is a standardized way of expressing how far away a specific data point is from the mean of a distribution, measured in terms of standard deviations.
We calculate the Z-score using the formula:\[Z = \frac{x - \mu}{\sigma}\]where \(x\) is the value of interest, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
It helps us understand how unusual or typical a data point is within a distribution. A Z-score:

• Greater than 0 indicates the data point is above the mean.
• Less than 0 indicates it is below the mean.
• Of 0 means it is exactly at the mean.

Z-scores are useful for comparing different data points across different normal distributions or understanding their relative standing in a specific distribution.

Standard deviation

The standard deviation is a measure that shows how much individual data points deviate from the mean of a dataset.
A smaller standard deviation indicates that the data points are closely clustered around the mean, while a larger standard deviation suggests that the data points are more spread out.
This metric is crucial because it provides insight into the variability or dispersion in a set of data:

• It is represented by the symbol \(\sigma\).
• Calculating standard deviation involves squaring the difference between each data point and the mean, taking the average of those squared differences, and then taking the square root.
• It is commonly used to interpret the spread around the mean in a dataset.
• In the context of the normal distribution, approximately 68% of data lies within one standard deviation from the mean, 95% within two, and 99.7% within three standard deviations.

Understanding standard deviation is essential for evaluating the consistency or variability of data.

Cumulative distribution function

The cumulative distribution function, or CDF, is a tool used to determine the probability that a random variable takes on a value less than or equal to a specific amount. It accumulates the probabilities of a random variable, providing a complete description of its distribution.
In a normal distribution, the CDF can be used to quickly find probabilities for ranges of values.
The function is defined for a random variable \(X\) as:\[F(x) = P(X \leq x)\]where \(P(X \leq x)\) is the probability that the variable is less than or equal to \(x\).
Key points about CDFs include:

• They start at 0 and increase to 1 as \(x\) increases.
• They are used to find the probability that a value falls within a particular range.
• The CDF of a normal distribution helps determine how extreme or common a value is relative to the rest of the dataset.

The cumulative distribution function smoothens the details from a probability distribution, showing how those probabilities accumulate cumulatively over the values of the variable.