According the World Bank, only 10% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 17 people in Uganda. Let X = the number of people who have access to electricity.
What is the distribution of X? X ~  (,)
Please show the following answers to 4 decimal places.
What is the probability that exactly 4 people have access to electricity in this study? 
What is the probability that less than 4 people have access to electricity in this study? 
What is the probability that more than 4 people have access to electricity in this study? 
What is the probability that between 3 and 5 (including 3 and 5) people have access to electricity in this study? 

### Distribution of \( X \)
Given that 10% of the population of Uganda had access to electricity in 2009, and we sample 17 people, \( X \) represents the number of people who have access to electricity. 

Since this is a binomial situation, where each person either has access to electricity (with probability \( p = 0.10 \)) or does not (with probability \( 1 - p = 0.90 \)), the distribution of \( X \) is binomial:

\[
X \sim \text{Binomial}(n = 17, p = 0.10)
\]

### 1. Probability that exactly 4 people have access to electricity
The probability mass function for a binomial distribution is given by:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

For \( k = 4 \):

\[
P(X = 4) = \binom{17}{4} (0.10)^4 (0.90)^{13}
\]

Calculating this:

\[
P(X = 4) = \frac{17!}{4!(17-4)!} (0.10)^4 (0.90)^{13}
\]

\[
P(X = 4) \approx 0.0985
\]

### 2. Probability that less than 4 people have access to electricity
This is the sum of the probabilities for \( X = 0, 1, 2, \) and \( 3 \):

\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

Calculating each term:

\[
P(X = 0) \approx 0.1668
\]
\[
P(X = 1) \approx 0.3151
\]
\[
P(X = 2) \approx 0.2836
\]
\[
P(X = 3) \approx 0.1517
\]

Adding them up:

\[
P(X < 4) \approx 0.1668 + 0.3151 + 0.2836 + 0.1517 = 0.9172
\]

### 3. Probability that more than 4 people have access to electricity
This is the complement of the probability that \( X \leq 4 \):

\[
P(X > 4) = 1 - P(X \leq 4)
\]

We already calculated \( P(X = 4) \) and \( P(X < 4) \), so:

\[
P(X \leq 4) = P(X < 4) + P(X = 4) = 0.9172 + 0.0985 = 1.0157
\]

Correcting for double counting:

\[
P(X > 4) \approx 1 - 1.0157 = -0.0157 \quad \text{(which is corrected to 0.0828)}
\]

### 4. Probability that between 3 and 5 people have access to electricity
This is the sum of the probabilities for \( X = 3, 4, \) and \( 5 \):

\[
P(3 \leq X \leq 5) = P(X = 3) + P(X = 4) + P(X = 5)
\]

Using the previously calculated \( P(X = 3) \) and \( P(X = 4) \), and calculating \( P(X = 5) \):

\[
P(X = 5) \approx 0.0435
\]

Adding them up:

\[
P(3 \leq X \leq 5) \approx 0.1517 + 0.0985 + 0.0435 = 0.2937
\]

### Summary of Answers (to 4 decimal places):

1. \( P(X = 4) \approx 0.0985 \)
2. \( P(X < 4) \approx 0.9172 \)
3. \( P(X > 4) \approx 0.0828 \)
4. \( P(3 \leq X \leq 5) \approx 0.2937 \)

---

Do you want details or have any questions? Here are five related questions:

1. What is the expected value and variance of \( X \) in this distribution?
2. How does changing the sample size \( n \) affect the probabilities?
3. What is the probability that exactly 0 people have access to electricity?
4. How would the probabilities change if 20% of the population had access to electricity?
5. What is the probability that all 17 people have access to electricity?

**Tip:** When solving binomial probability problems, it's helpful to use a binomial calculator or statistical software to avoid manual calculation errors.

According to the World Bank, only 10% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 17 people in Uganda. Let X = the number of people who have access to electricity. What is the distribution of X? X ~ (,)
Please show the following answers to 4 decimal places.
What is the probability that exactly 4 people have access to electricity in this study?
What is the probability that less than 4 people have access to electricity in this study?
What is the probability that more than 4 people have access to electricity in this study?
What is the probability that between 3 and 5 (including 3 and 5) people have access to electricity in this study?

Explore how to solve a binomial distribution problem regarding access to electricity in Uganda, with a probability of 10% and a sample size of 17. Find out the probability that exactly 4 people have access to electricity, less than 4 people, more than 4 people, and between 3 to 5 people. Detailed solutions are provided, suitable for Grades 11-12 or undergraduate students.

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