A real estate agent has 15
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 no more than 2
properties in one week. Round your answer to four decimal places.

This problem can be modeled using the **binomial distribution** because it involves repeated independent trials, each with two possible outcomes (selling or not selling a property).

The binomial distribution formula is:

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

Where:
- \( n = 15 \) (number of properties),
- \( p = 0.40 \) (probability of selling one property),
- \( k \) is the number of properties sold,
- \( \binom{n}{k} \) is the binomial coefficient, or "n choose k", which gives the number of ways to choose \( k \) successes out of \( n \) trials.

We need to find the probability of selling no more than 2 properties, i.e., \( P(X \leq 2) \).

So, we calculate \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \), and sum them:

### 1. \( P(X = 0) \):
\[
P(X = 0) = \binom{15}{0} (0.40)^0 (0.60)^{15} = 1 \times 1 \times (0.60)^{15} = 0.60^{15}
\]

### 2. \( P(X = 1) \):
\[
P(X = 1) = \binom{15}{1} (0.40)^1 (0.60)^{14} = 15 \times 0.40 \times (0.60)^{14}
\]

### 3. \( P(X = 2) \):
\[
P(X = 2) = \binom{15}{2} (0.40)^2 (0.60)^{13} = \frac{15 \times 14}{2} \times (0.40)^2 \times (0.60)^{13}
\]

Now, summing these results gives the total probability \( P(X \leq 2) \). Let me compute this.

### Final result:
\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]The probability of selling no more than 2 properties in one week is approximately **0.0271** (rounded to four decimal places).

Would you like more details or have any questions?

Here are five related questions to further explore this topic:
1. What is the probability of selling exactly 3 properties?
2. What is the probability of selling more than 2 properties?
3. How does the probability change if the chance of selling a property increases to 50%?
4. What is the expected number of properties sold in a week?
5. How would the variance and standard deviation for the number of properties sold be calculated?

**Tip:** The sum of all probabilities for any binomial distribution always equals 1, as they cover all possible outcomes.

A real estate agent has 15 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 no more than 2 properties in one week. Round your answer to four decimal places.

Explore the probability of selling no more than 2 properties out of 15, with a 40% chance of selling each. This problem uses the binomial distribution formula and is suitable for students in advanced high school or college-level probability courses.

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