A real estate agent has 17
 properties that she shows. She feels that there is a 30%
 chance of selling any one property during a week. The chance of selling any one property is independent of selling another property. Compute the probability of selling more than 4
 properties in one week. Round your answer to four decimal places.

This problem involves calculating the probability of selling more than 4 properties out of 17, where the probability of selling each property is \( 30\% \), or \( p = 0.30 \), and the probability of not selling a property is \( 1 - p = 0.70 \).

This situation can be modeled using the **binomial distribution**, which is used when we have independent events with two possible outcomes (success or failure) over a fixed number of trials. The binomial probability mass function is given by:

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

where:
- \( n = 17 \) (number of trials, or properties),
- \( p = 0.30 \) (probability of success, selling a property),
- \( k \) (number of successes, or properties sold).

### Step 1: Total probability of selling more than 4 properties
We need to compute \( P(X > 4) \), which is the complement of the probability of selling 4 or fewer properties:

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

### Step 2: Compute \( P(X \leq 4) \)
The probability \( P(X \leq 4) \) is the sum of the probabilities of selling 0, 1, 2, 3, or 4 properties:

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

We will calculate each of these probabilities using the binomial formula, then sum them and subtract from 1 to get the desired result.

### Step 3: Perform calculations
Let's proceed to calculate each probability term and then sum them up.

The probability of selling more than 4 properties in one week is approximately \( 0.6113 \) or 61.13%, rounded to four decimal places.

Would you like more details or have any questions about this? 

Here are 5 related questions for further exploration:

1. What is the probability of selling exactly 4 properties in a week?
2. How does the probability change if the chance of selling a property increases to 40%?
3. What is the expected number of properties sold during the week?
4. How would the calculation change if the agent had only 10 properties to sell?
5. What is the variance and standard deviation of the number of properties sold?

**Tip:** In a binomial distribution, the sum of probabilities for all possible outcomes always equals 1.

A real estate agent has 17 properties that she shows. She feels that there is a 30% chance of selling any one property during a week. The chance of selling any one property is independent of selling another property. Compute the probability of selling more than 4 properties in one week. Round your answer to four decimal places.

Explore how to calculate the probability of selling more than 4 properties out of 17, where the probability of selling each is 30%. This binomial distribution problem includes detailed steps on using probability theory and binomial formulas. Suitable for Grades 10-12.

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