This problem can be modeled using the binomial distribution, which is appropriate when there are a fixed number of independent trials, each with the same probability of success. Here’s how to break it down:

Given:

• Number of properties (n) = 15
• Probability of selling a property (p) = 0.60
• The chance of selling more than 5 properties means we want to find $P(X > 5)$, where $X$ represents the number of properties sold in a week.
• The probability of not selling a property is $1 - p = 0.40$.

We can compute the probability of selling more than 5 properties by finding the complementary probability $P(X \leq 5)$ (i.e., selling 5 or fewer properties), and then subtracting that from 1:

$P(X > 5) = 1 - P(X \leq 5)$

Step 1: Use the Binomial CDF

The probability of selling exactly $k$ properties in a week follows the binomial distribution:

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

For $P(X \leq 5)$, we sum the probabilities for $k = 0$ through $k = 5$:

$P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)$

Step 2: Calculate

Now let’s compute the cumulative probability $P(X \leq 5)$. For binomial distributions, this is often done using a binomial cumulative distribution function (CDF), which is available in statistical software or scientific calculators.

Using this approach, we find that:

$P(X \leq 5) \approx 0.1798$

Step 3: Compute $P(X > 5)$

Now, we subtract this result from 1 to find the probability of selling more than 5 properties:

$P(X > 5) = 1 - 0.1798 = 0.8202$

Final Answer:

The probability of selling more than 5 properties in one week is approximately 0.8202.

Would you like a more detailed breakdown of the calculations, or do you have any further questions?

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Here are some related questions to expand your understanding:

1. How would the probability change if the chance of selling any one property increased to 80%?
2. How does the binomial distribution compare to the normal distribution in this type of problem?
3. How can you calculate the expected number of properties sold?
4. How would you approach the problem if the trials were dependent on each other?
5. What is the variance and standard deviation of the number of properties sold?

Tip: For binomial distributions, calculating cumulative probabilities (like $P(X \leq k)$) is often done more efficiently using a calculator or software, especially for large numbers of trials like 15.