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Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. According to The World Bank, only 9% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 150 people in Uganda. Let \(X\) = the number of people who have access to electricity. a. What is the probability distribution for \(X\)? b. Using the formulas, calculate the mean and standard deviation of \(X\). c. Use your calculator to find the probability that 15 people in the sample have access to electricity. d. Find the probability that at most ten people in the sample have access to electricity. e. Find the probability that more than 25 people in the sample have access to electricity.

Step by step solution

01

Identify the type of probability distribution for X

The number of individuals in a sample having a certain trait (such as electricity access) typically follows a binomial distribution. This is appropriate here, as there are two outcomes (having or not having electricity), a fixed number of trials (150), and a consistent probability across trials. Thus, \(X\) is a binomial random variable with parameters \(n = 150\), \(p = 0.09\).

02

Calculate the mean of the distribution

The mean (expected value) \(\mu\) of a binomial distribution is given by \(\mu = n \times p\). Substituting in the given values, \(\mu = 150 \times 0.09 = 13.5\). This means, on average, 13.5 people in the sample will have electricity access.

03

Calculate the standard deviation of the distribution

The standard deviation \(\sigma\) for a binomial variable is given by \(\sigma = \sqrt{n \times p \times (1-p)}\). Substituting the values, \(\sigma = \sqrt{150 \times 0.09 \times 0.91} \approx 3.54\).

04

Calculate the probability that exactly 15 people have electricity

Use the binomial probability formula, \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\), where \(k = 15\). Substituting the values: \(P(X=15) = \binom{150}{15} (0.09)^{15} (0.91)^{135}\). This calculation usually requires a calculator or software due to its complexity.

05

Calculate the probability that at most 10 people have electricity

The probability that \(X\) is at most 10 is \(P(X \leq 10)\), which is the sum of the probabilities from \(X = 0\) to \(X = 10\). Use the cumulative distribution function (CDF) for the binomial distribution: \(P(X \leq 10)\). This also usually requires a calculator or statistical software.

06

Calculate the probability that more than 25 people have electricity

This is calculated as \(P(X > 25) = 1 - P(X \leq 25)\). First, find \(P(X \leq 25)\) using the CDF for the binomial distribution and then subtract from 1. Use a calculator or software to compute this efficiently.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

In probability theory, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. Specifically, a binomial probability distribution is used when an experiment meets certain criteria:

• There are a fixed number of trials.
• Each trial has two possible outcomes, often termed as "success" and "failure".
• There is a constant probability of success in each trial.

In our example, the number of people in Uganda who have access to electricity follows a binomial distribution. Here, "success" is defined as having access to electricity. With 150 trials (or individuals), each having a 9% chance of being a success, the distribution of the random variable, denoted as X, is determined by these parameters.

Mean and Standard Deviation

Mean and standard deviation are two critical properties of a distribution that help us understand the data. The mean, or expected value equation is expressed as \(\mu = n \times p\);this tells us the average outcome we expect from the experiment. For our Ugandan electricity access example, the mean is 13.5, meaning that out of 150 people, about 13.5 are expected to have electricity.

The standard deviation measures the dispersion or spread of the distribution around the mean. In a binomial distribution, the standard deviation \(\sigma\) is calculated as\(\sigma = \sqrt{n \times p \times (1-p)}\).This provides a sense of the variability, or how much the actual data might deviate from the mean. For our binomial experiment, the standard deviation is approximately 3.54, indicating that while we expect around 13.5 "successes" (people with electricity), the actual number may vary by about 3.54 around this mean.

Binomial Probability Formula

The binomial probability formula is used to determine the probability of observing a specific number of "successes" in a given number of trials. The formula is:\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here, \(\binom{n}{k}\) is a binomial coefficient that calculates the number of ways to choose \(k\) successes out of \(n\) trials. In our case, if we want to find the probability that exactly 15 out of 150 individuals have access to electricity, \(k = 15\). Substituting the values into the formula provides the probability of this exact scenario. Calculating this manually can be complex, hence typically, calculators or statistical software are used for practical computation.

Cumulative Distribution Function

The cumulative distribution function (CDF) aggregates probabilities up to a given point and is crucial for calculating probabilities over a range of values. For a binomial distribution, the CDF tells us the probability that the random variable \(X\) is less than or equal to a particular value. For instance, if we're interested in finding the probability that at most 10 people have electricity, we would use \(P(X \leq 10)\).

The CDF can also help calculate the probability of more than a certain number of successes by using the relation
\[ P(X > k) = 1 - P(X \leq k) \]This is particularly useful when determining probabilities such as more than 25 people having access. Tools like statistical calculators or software can handle these calculations efficiently to give swift and accurate results.