A real estate agent has 12
 properties that she shows. She feels that there is a 40%
 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 3
 properties in one week. Round your answer to four decimal places.



To solve this problem, we can use the binomial distribution formula. The binomial distribution is used because the problem involves a fixed number of trials (12 properties), each with two possible outcomes (sold or not sold), and each trial is independent with the same probability of success.

The formula for the binomial probability is:

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

Where:
- \( n \) is the number of trials (properties), which is 12.
- \( k \) is the number of successes (properties sold).
- \( p \) is the probability of success on each trial, which is 0.40.
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

We are asked to compute the probability of selling more than 3 properties. This means we need to find:

\[
P(X > 3) = 1 - P(X \leq 3)
\]

First, we calculate \( P(X \leq 3) \), which is the sum of the probabilities of selling 0, 1, 2, or 3 properties.

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

Let's calculate each probability and then sum them up.

### Step 1: Calculate \( P(X = 0) \):

\[
P(X = 0) = \binom{12}{0} \cdot (0.4)^0 \cdot (0.6)^{12} 
\]

### Step 2: Calculate \( P(X = 1) \):

\[
P(X = 1) = \binom{12}{1} \cdot (0.4)^1 \cdot (0.6)^{11} 
\]

### Step 3: Calculate \( P(X = 2) \):

\[
P(X = 2) = \binom{12}{2} \cdot (0.4)^2 \cdot (0.6)^{10} 
\]

### Step 4: Calculate \( P(X = 3) \):

\[
P(X = 3) = \binom{12}{3} \cdot (0.4)^3 \cdot (0.6)^{9} 
\]

### Step 5: Sum the probabilities:

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

Finally, subtract this sum from 1 to find \( P(X > 3) \).

Let’s calculate this.The probability of selling more than 3 properties in one week is **0.7747** (rounded to four decimal places).

Would you like further details or have any other questions? Here are some related questions you might find interesting:

1. What is the probability of selling exactly 5 properties?
2. How does the probability change if the probability of selling each property increases to 50%?
3. What is the expected number of properties sold in a week?
4. What is the standard deviation of the number of properties sold in a week?
5. How does the probability distribution look for different numbers of properties sold?

**Tip:** When dealing with binomial distributions, understanding the relationship between probability, number of trials, and outcomes can help in predicting scenarios effectively.

A real estate agent has 12 properties that she shows. She feels that there is a 40% 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 3 properties in one week. Round your answer to four decimal places.

Calculate the probability of selling more than 3 properties in one week using the binomial distribution formula. This problem involves 12 trials, with a 40% chance of success on each. Detailed step-by-step solution included. Suitable for Grades 10-12.

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