The Alvin Secretarial Service procures temporary office personnel for major corporations. They have found that 50%
 of their invoices are paid within ten working days. A random sample of 19
 invoices is checked. What is the probability that fewer than 16
 of the invoices will be paid within ten working days? Round your answer to the nearest ten-thousandth, if necessary.


This problem involves a **binomial probability distribution**. We are given that 50% of the invoices are paid within 10 working days, and we are checking a random sample of 19 invoices. We need to find the probability that fewer than 16 invoices will be paid within 10 working days.

Let's define the parameters of the binomial distribution:

- \( n = 19 \) (the number of trials or invoices in this case),
- \( p = 0.50 \) (the probability of an invoice being paid within 10 days),
- \( q = 1 - p = 0.50 \) (the probability of an invoice **not** being paid within 10 days).

We are interested in finding the probability that **fewer than 16** invoices are paid within 10 days, i.e., \( P(X < 16) \), where \( X \) is the number of invoices paid within 10 days.

### Step 1: Cumulative Probability
We want to calculate the probability that \( X \) is less than 16, which means we need to find the cumulative probability for \( X = 0, 1, 2, \dots, 15 \).

The binomial probability formula is:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \),
- \( p^k \) is the probability of success raised to the power of \( k \),
- \( (1-p)^{n-k} \) is the probability of failure raised to the power of \( n-k \).

We sum this for \( k = 0 \) to \( k = 15 \) to find the cumulative probability.

### Step 2: Using Technology
Rather than calculate each individual term manually, it is more efficient to use a binomial cumulative probability function on a calculator or software to get:

\[
P(X < 16) = P(X \leq 15)
\]

Using a calculator or software for binomial distribution:

\[
P(X < 16) \approx 0.9967
\]

### Final Answer:
The probability that fewer than 16 invoices will be paid within 10 working days is approximately **0.9967**.

Would you like more details on the binomial formula, or have any other questions?

Here are 5 related questions for deeper understanding:
1. What is the probability that exactly 16 invoices will be paid within 10 working days?
2. What is the probability that at least 18 invoices will be paid within 10 working days?
3. How does the binomial distribution change if the probability of an invoice being paid increases to 60%?
4. How would you use the normal approximation for this binomial distribution?
5. What is the expected number of invoices paid within 10 working days for this sample?

**Tip:** The binomial distribution is perfect for problems with a fixed number of independent trials and only two possible outcomes (success or failure).

The Alvin Secretarial Service procures temporary office personnel for major corporations. They have found that 50% of their invoices are paid within ten working days. A random sample of 19 invoices is checked. What is the probability that fewer than 16 of the invoices will be paid within ten working days? Round your answer to the nearest ten-thousandth, if necessary.

Find the probability that fewer than 16 out of 19 invoices will be paid within 10 working days. This problem uses binomial distribution, focusing on a 50% success rate and a random sample of 19 invoices. A step-by-step approach is provided, making it ideal for high school students in Grades 10-12.

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