A real estate agent has 18
properties that she shows. She feels that there is a 50%
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 4
properties in one week. Round your answer to four decimal places.

To solve this problem, we can use the **binomial distribution** formula, as the situation involves a fixed number of independent trials (selling properties) with only two possible outcomes (selling a property or not selling it).

### Given:
- Number of trials (\(n\)) = 18 properties
- Probability of success in each trial (\(p\)) = 0.5 (50% chance of selling any one property)
- Probability of failure (\(q\)) = \(1 - p = 0.5\)
- We need to find the probability of selling no more than 4 properties, i.e., \(P(X \leq 4)\).

The probability mass function (PMF) for a binomial distribution is given by:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where \( \binom{n}{k} \) is the binomial coefficient, calculated as:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

### Calculation:
We need to calculate the probability for \( X = 0, 1, 2, 3, \) and \( 4 \) and then sum these probabilities:

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

Let's calculate this using the binomial formula.

1. **For \(X = 0\)**:
   \[
   P(X = 0) = \binom{18}{0} (0.5)^0 (0.5)^{18} = 1 \times (0.5)^{18} = \frac{1}{2^{18}}
   \]

2. **For \(X = 1\)**:
   \[
   P(X = 1) = \binom{18}{1} (0.5)^1 (0.5)^{17} = 18 \times (0.5)^{18}
   \]

3. **For \(X = 2\)**:
   \[
   P(X = 2) = \binom{18}{2} (0.5)^2 (0.5)^{16} = \frac{18 \times 17}{2} \times (0.5)^{18}
   \]

4. **For \(X = 3\)**:
   \[
   P(X = 3) = \binom{18}{3} (0.5)^3 (0.5)^{15} = \frac{18 \times 17 \times 16}{6} \times (0.5)^{18}
   \]

5. **For \(X = 4\)**:
   \[
   P(X = 4) = \binom{18}{4} (0.5)^4 (0.5)^{14} = \frac{18 \times 17 \times 16 \times 15}{24} \times (0.5)^{18}
   \]

Summing these will give us \( P(X \leq 4) \).

I'll calculate this for you.The probability of selling no more than 4 properties in one week is approximately **0.0154** (rounded to four decimal places).

Would you like any more details or have any questions? Here are five related questions to expand on this topic:

1. How would the probability change if the agent had 20 properties instead of 18?
2. What is the probability of selling exactly 5 properties in one week?
3. How does the probability change if the chance of selling a property increases to 60%?
4. Can we use a normal approximation for this binomial distribution? If so, how accurate would it be?
5. How do we calculate the expected number of properties sold in a week?

**Tip:** The binomial distribution is useful when dealing with a fixed number of independent trials, each with the same probability of success.

A real estate agent has 18 properties that she shows. She feels that there is a 50% 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 4 properties in one week. Round your answer to four decimal places.

This problem explores the use of the binomial distribution to compute the probability of selling no more than 4 out of 18 properties, given a 50% chance of selling each. Learn step-by-step how to apply the binomial formula, useful for high school students studying probability.

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