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.