This problem can be modeled using the binomial distribution. The situation is as follows:

• The number of trials $n$ is 18 (the number of properties shown).
• The probability of success on a single trial $p$ is 0.40 (the chance of selling a property).
• We are interested in the probability of selling more than 2 properties, which means we want to find $P(X > 2)$, where $X$ is the number of properties sold.

The probability mass function (PMF) of a binomial distribution is given by:

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

Where:

• $n = 18$
• $p = 0.40$
• $k$ is the number of properties sold (we'll calculate $P(X > 2)$).

We need to compute $P(X > 2)$, which is the complement of $P(X \leq 2)$:

$P(X > 2) = 1 - P(X \leq 2)$

Where:

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

Now, let's compute these individual probabilities:

Step 1: Calculate $P(X = 0)$, $P(X = 1)$, and $P(X = 2)$

1. $P(X = 0) = \binom{18}{0} (0.40)^0 (0.60)^{18}$
2. $P(X = 1) = \binom{18}{1} (0.40)^1 (0.60)^{17}$
3. $P(X = 2) = \binom{18}{2} (0.40)^2 (0.60)^{16}$

Step 2: Use the complement rule to find $P(X > 2)$

Once the individual probabilities are calculated, we subtract their sum from 1 to get the desired probability.

Let me calculate these values for you. math

Given values

n = 18 # Number of properties shown p = 0.40 # Probability of selling one property q = 1 - p # Probability of not selling a property

Binomial probability function

def binomial_prob(n, k, p): return math.comb(n, k) * (p ** k) * (q ** (n - k))

Calculate P(X = 0), P(X = 1), and P(X = 2)

P_X_0 = binomial_prob(n, 0, p) P_X_1 = binomial_prob(n, 1, p) P_X_2 = binomial_prob(n, 2, p)

Calculate P(X <= 2)

P_X_leq_2 = P_X_0 + P_X_1 + P_X_2

Calculate P(X > 2)

P_X_gt_2 = 1 - P_X_leq_2 P_X_gt_2_rounded = round(P_X_gt_2, 4)

P_X_gt_2_roundedThe probability of selling more than 2 properties in one week is approximately 0.9918.

Let me know if you'd like further details or have any other questions!

Additional Questions:

1. How do the binomial distribution and normal distribution compare for this type of problem?
2. What is the probability of selling exactly 5 properties in a week?
3. How would the probability change if the chance of selling a property was 50% instead of 40%?
4. What is the expected number of properties sold in a week?
5. How can you use the Poisson distribution to approximate this problem if the number of trials is very large?

Tip:

If you're working with binomial probabilities and the number of trials is large, you can often approximate the binomial distribution with a normal distribution for easier calculations.