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Chapter 5: Problem 27

Records show that \(30 \%\) of all patients admitted to a medical clinic fail to pay their bills and that eventually the bills are forgiven. Suppose \(n=4\) new patients represent a random selection from the large set of prospective patients served by the clinic. Find these probabilities: a. All the patients' bills will eventually have to be forgiven. b. One will have to be forgiven. c. None will have to be forgiven.

Short Answer

Expert verified
Answer: The probabilities are a) 0.0081, b) 0.4116, and c) 0.2401.

Step by step solution

01

Use the binomial formula

We will use the formula \(P(x) = \binom{n}{x} p^x (1-p)^{(n-x)}\) with \(n=4\), \(p=0.30\), and \(x=4\).
02

Calculate the binomial coefficient

Calculate the binomial coefficient \(\binom{4}{4} = \frac{4!}{4!(4-4)!} = 1\).
03

Compute the probability

Now, compute the probability: \(P(4) = \binom{4}{4} (0.30)^4 (0.70)^0 = 1 \cdot 0.0081 \cdot 1 = 0.0081\). The probability that all patients' bills will eventually have to be forgiven is \(0.0081\). b. One will have to be forgiven.
04

Use the binomial formula

We will use the same formula as before, but this time with \(x=1\).
05

Calculate the binomial coefficient

Calculate the binomial coefficient \(\binom{4}{1} = \frac{4!}{1!(4-1)!} = \frac{4!}{3!} = 4\).
06

Compute the probability

Compute the probability: \(P(1) = \binom{4}{1} (0.30)^1 (0.70)^3 = 4 \cdot 0.30 \cdot 0.343 = 0.4116\). The probability that only one patient's bill will have to be forgiven is \(0.4116\). c. None will have to be forgiven.
07

Use the binomial formula

We will again use the same formula, but this time with \(x=0\).
08

Calculate the binomial coefficient

Calculate the binomial coefficient \(\binom{4}{0} = \frac{4!}{0!(4-0)!} = 1\).
09

Compute the probability

Compute the probability: \(P(0) = \binom{4}{0} (0.30)^0 (0.70)^4 = 1 \cdot 1 \cdot 0.2401 = 0.2401\). The probability that none of the patients' bills will have to be forgiven is \(0.2401\). In conclusion, the probabilities are: a. 0.0081 for all patients' bills being forgiven. b. 0.4116 for one patient's bill being forgiven. c. 0.2401 for none of the patients' bills being forgiven.

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