A real estate agent has 18
 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 less than 2
 properties in one week. Round your answer to four decimal places.

This problem follows a **binomial distribution** because each property has two possible outcomes: it is either sold or not sold. The parameters for a binomial distribution are:

- \( n = 18 \) (the number of properties),
- \( p = 0.40 \) (the probability of selling a property),
- \( q = 1 - p = 0.60 \) (the probability of not selling a property).

We are tasked with finding the probability of selling less than 2 properties, i.e., \( P(X < 2) \). This is the sum of the probabilities of selling 0 or 1 property:

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

The formula for the binomial probability mass function is:

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

### 1. Probability of selling 0 properties (\( P(X = 0) \)):

\[
P(X = 0) = \binom{18}{0} (0.40)^0 (0.60)^{18}
\]

### 2. Probability of selling 1 property (\( P(X = 1) \)):

\[
P(X = 1) = \binom{18}{1} (0.40)^1 (0.60)^{17}
\]

Let me compute these values.The probability of selling less than 2 properties in one week is approximately **0.0013** (rounded to four decimal places).

Would you like further details or have any questions? Here are 5 related questions to expand on the topic:

1. How do you calculate probabilities for more than 2 properties?
2. What happens to the probability if the number of properties increases?
3. How is the binomial distribution different from other probability distributions?
4. What is the expected number of properties sold in a week?
5. How can we use the cumulative probability function to simplify this problem?

**Tip:** In binomial distributions, the expected value is calculated as \( E(X) = n \cdot p \).

A real estate agent has 18 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 less than 2 properties in one week. Round your answer to four decimal places.

Calculate the probability of selling less than 2 properties out of 18, using binomial distribution. This problem explores key concepts in probability, including the binomial probability theorem, and is suitable for students in Grades 10-12.

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