Question

Refer to Exercise 5.27 where \(30 \%\) of all admitted patients fail to pay their bills and the debts are eventually forgiven. Suppose that the clinic treats 2000 different patients over a period of 1 year, and let \(x\) be the number of forgiven debts. a. What is the mean (expected) number of debts that have to be forgiven? b. Find the variance and standard deviation of \(x\). c. What can you say about the probability that \(x\) will exceed \(700 ?\) (HINT: Use the values of \(\mu\) and \(\sigma\), along with Tchebysheff's Theorem.)

Short answer

Question: Determine the probability that the number of forgiven debts will exceed 700 patients. Answer: The probability that the number of forgiven debts will exceed 700 patients is less than or equal to 4.19%.

Step-by-step solution

Find the mean number of forgiven debts

To calculate the mean number of debts that have to be forgiven, we multiply the probability of a debt being forgiven (30%) by the total number of patients treated (2000). In this case, $$ \mu = (0.30)(2000) = 600 $$ So, the mean or expected number of debts that have to be forgiven is 600.

Find the variance of \(x\)

The variance of \(x\) can be calculated using the formula: $$ \sigma^2 = n*p*q, $$ where \(n\) is the total number of patients treated, \(p\) is the probability that a debt is forgiven, and \(q\) is the probability that a debt is not forgiven. In this case, \(n=2000\), \(p=0.3\), and \(q=0.7\). So, $$ \sigma^2 = (2000)(0.3)(0.7) = 420. $$ The variance of the number of forgiven debts is 420.

Find the standard deviation of \(x\)

To find the standard deviation \(\sigma\) of \(x\), take the square root of the variance. $$ \sigma = \sqrt{420} \approx 20.49 $$ So, the standard deviation of the number of forgiven debts is approximately 20.49.

Use Tchebysheff's Theorem to find the probability that \(x\) will exceed 700

To calculate the probability that the number of forgiven debts (\(x\)) will exceed 700, we first find the number of standard deviations (k) from the mean (600) the value 700 falls on. In this case, we have: $$ k = \frac{700 - 600}{20.49} \approx 4.88 $$ Using Tchebysheff's Theorem, we know that: $$ P(k \cdot \sigma) \geq 1 - \frac{1}{k^2} $$ Substitute \(k \approx 4.88\) into the formula: $$ P(4.88 \cdot 20.49) \geq 1-\frac{1}{4.88^2}, $$ $$ P(99.89) \geq 1-\frac{1}{23.85}, $$ $$ P(x \ge 700) \le \frac{1}{23.85} \approx 0.0419. $$ So, we can say that the probability that the number of forgiven debts, \(x\), will exceed 700 is less than or equal to 4.19%.

Key concepts

Mean and Expected Value

The concept of mean or expected value is crucial in probability theory. It represents the average outcome you'd expect from a probability experiment if it were repeated many times. To calculate the mean for a given scenario, multiply each possible outcome by the probability of that outcome occurring and then sum these products.

In the context of our exercise, we want to find the expected number of debts forgiven at a clinic. With the probability of a debt being forgiven set at 30%, and 2000 patients treated, we calculate the expected value by multiplying 2000 by 0.30 (the probability). Thus, the mean, \(\mu, is \(\mu = (0.30)(2000) = 600\)\).

This result tells us that on average, out of 2000 patients treated in a year, 600 are expected to result in forgiven debts. Expected value provides a central figure around which random variables tend to gather.

Variance and Standard Deviation

Variance and standard deviation measure the spread of a data set. Variance quantifies how far a set of numbers are from their average value, while the standard deviation gives you a sense of how much variance exists in a simpler, more interpretable form.

To find the variance of the number of forgiven debts, use the formula \(\sigma^2 = n \, p \, q\), where \(n\) is the number of trials (2000 patients), \(p\) is the probability of success (a debt forgiven), and \(q\) is the probability of failure (a debt not forgiven). Substituting the values, we have \(\sigma^2 = (2000)(0.3)(0.7) = 420\). So, the variance of the number of forgiven debts is 420.

Standard deviation provides an easier way to interpret the variance. It's the square root of the variance, \(\sigma = \sqrt{420} \approx 20.49\). This value indicates how much the number of forgiven debts will typically deviate from the mean on average. Smaller standard deviations indicate that data points are clustered closely around the mean, while larger values suggest more dispersion.

Tchebysheff's Theorem

Tchebysheff's theorem provides a way to make probabilistic statements about a random variable bound by knowledge of its mean and standard deviation, even without knowing its exact distribution. The theorem states that for any k>1, at least \(1 - \frac{1}{k^2}\) of the data will fall within \(k\) standard deviations of the mean.

Using Tchebysheff's theorem, we assess the probability of exceeding a certain number of forgiven debts. To find where 700 falls compared to the mean of 600, calculate how many standard deviations 700 is from 600: \(k = \frac{700 - 600}{20.49} \approx 4.88\). Tchebysheff's theorem then implies that at least \(1 - \frac{1}{4.88^2}\) of the data falls within these bounds: \(P(x \ge 700) \le \frac{1}{23.85} \approx 0.0419\).

So, with this theorem, we can say there's less than a 4.19% chance that 700 or more debts are forgiven. This highlights Tchebysheff's utility in setting constraints when sample distribution details are unavailable.