This problem involves the binomial distribution, where we can model the probability of success (an invoice being paid within ten working days) and failure (not being paid within ten working days). The binomial distribution is appropriate here because each invoice has only two possible outcomes (paid within ten days or not), and the number of invoices checked is fixed at 18.

Given:

• Probability of success, $p = 0.50$ (50% of invoices are paid within 10 days)
• Number of trials, $n = 18$ (random sample of 18 invoices)
• We need to find the probability that less than 15 invoices are paid within 10 days, i.e., $P(X < 15)$.

Binomial Probability Formula:

The probability mass function for the binomial distribution is:

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

Where:

• $\binom{n}{k}$ is the binomial coefficient
• $p$ is the probability of success
• $1 - p$ is the probability of failure
• $k$ is the number of successes (invoices paid within ten days)
• $n$ is the total number of trials

We need to compute the cumulative probability $P(X < 15)$, which is the sum of probabilities from 0 to 14:

$P(X < 15) = P(X = 0) + P(X = 1) + \cdots + P(X = 14)$

However, using a binomial cumulative distribution function (CDF) simplifies this.

Using CDF:

We calculate:

$P(X < 15) = P(X \leq 14)$

This can be computed directly using a binomial cumulative distribution function from tables, a calculator, or statistical software.

Computation:

Using the binomial CDF, we find $P(X \leq 14)$ for $n = 18$ and $p = 0.50$:

$P(X < 15) = 0.9969$

Thus, the probability that fewer than 15 invoices will be paid within 10 working days is approximately 0.9969.

Final Answer:

The probability is 0.9969 (rounded to four decimal places).

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Do you want more details on this solution or have any questions? Here are five related questions:

1. How do we derive the binomial coefficient in the probability formula?
2. What is the probability that exactly 15 invoices will be paid on time?
3. How do we use the binomial distribution to model different probabilities?
4. How can the binomial distribution be approximated with a normal distribution?
5. What is the expected number of invoices to be paid within 10 days?

Tip: When solving binomial problems, using the cumulative distribution function (CDF) simplifies calculations for probabilities involving ranges of outcomes.