Question 1185670: Medical billing errors and fraud are on the rise. According to the MBAA website, 8 out of 10 times, the medical bills that you get are not right. If a sample of 10 medical bills is selected, what is the probability that
a. 0 medical bills will contain errors?
b. Exactly 5 medical bills will contain errors?
c. More than 5 medical bills will contain errors?
d. What are the mean and standard deviation of the probability distribution
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem using the binomial probability distribution:
**Understanding the Problem**
This is a binomial probability problem because:
* There are a fixed number of trials (n = 10 medical bills).
* Each trial is independent.
* There are only two outcomes (the bill contains errors or it doesn't).
* The probability of success (a bill containing errors) is constant (p = 0.8).
* The probability of failure (a bill not containing errors) is q = 1 - p = 0.2.
The binomial probability formula is:
P(x) = (nCx) * p^x * q^(n-x)
where:
* P(x) is the probability of x successes
* n is the number of trials
* x is the number of successes
* p is the probability of success
* q is the probability of failure
* nCx is the binomial coefficient, calculated as n! / (x! * (n-x)!)
**a. 0 medical bills will contain errors:**
P(0) = (10C0) * (0.8)^0 * (0.2)^10
P(0) = 1 * 1 * (0.2)^10
P(0) ≈ 1.024 x 10^-7
**b. Exactly 5 medical bills will contain errors:**
P(5) = (10C5) * (0.8)^5 * (0.2)^5
P(5) = 252 * 0.32768 * 0.00032
P(5) ≈ 0.0264
**c. More than 5 medical bills will contain errors:**
This means we need to find the probability of 6, 7, 8, 9, or 10 bills containing errors and add them together. It's often easier to use the complement rule:
P(x > 5) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4) + P(5)]
Calculating each of these probabilities and summing them, then subtracting from 1, will give you P(x > 5). Due to the small probability of P(0) and P(1), we can ignore the results of P(0), P(1)
P(2) = (10C2) * (0.8)^2 * (0.2)^8 ≈ 0.00088
P(3) = (10C3) * (0.8)^3 * (0.2)^7 ≈ 0.0088
P(4) = (10C4) * (0.8)^4 * (0.2)^6 ≈ 0.088
P(5) ≈ 0.264
P(x > 5) = 1 - [0.00088 + 0.0088 + 0.088 + 0.264] ≈ 1 - 0.36168 ≈ 0.638
**d. Mean and Standard Deviation:**
For a binomial distribution:
* Mean (μ) = n * p = 10 * 0.8 = 8
* Standard deviation (σ) = sqrt(n * p * q) = sqrt(10 * 0.8 * 0.2) = sqrt(1.6) ≈ 1.265
Answer by ikleyn(52810)  (Show Source):
You can put this solution on YOUR website! .
Medical billing errors and fraud are on the rise. According to the MBAA website, 8 out of 10 times,
the medical bills that you get are not right. If a sample of 10 medical bills is selected, what is the probability that
a. 0 medical bills will contain errors?
b. Exactly 5 medical bills will contain errors?
c. More than 5 medical bills will contain errors?
d. What are the mean and standard deviation of the probability distribution
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In the post by @CPhill, all calculations in part (c) are incorrect.
I came to provide correct calculations.
**c. More than 5 medical bills will contain errors:**
This means we need to find the probability of 6, 7, 8, 9, or 10 bills containing errors and add them together.
It's often easier to use the complement rule:
P(x > 5) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4) + P(5)]
Calculating each of these probabilities and summing them, then subtracting from 1, will give you P(x > 5).
Due to the small probability of P(0) and P(1), we can ignore the results of P(0), P(1)
P(2) = (10C2) * (0.8)^2 * (0.2)^8 = 45 * 0.8^2 * 0.2^8 ≈ 7.3728E-05
P(3) = (10C3) * (0.8)^3 * (0.2)^7 = 120 * 0.8^3 * 0.2^7 ≈ 0.000786432
P(4) = (10C4) * (0.8)^4 * (0.2)^6 = 210 * 0.8^4 * 0.2^6 ≈ 0.005505024
P(5) = (10C5) * (0.8)^5 * (0.2)^5 = 252 * 0.8^5 * 0.2^5 ≈ 0.026424115
P(x > 5) = 1 - (7.3728E-05 + 0.000786432 + 0.005505024 + 0.264] ≈ 0.9672. correct ANSWER to (c)
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If it is an Artificial Intelligence, then how it may happen
that elementary arithmetic calculations are incorrect ?
I do not believe to my eyes . . .
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