SOLUTION: 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

Algebra.Com
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(52814)   (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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        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)

------------------------------------------

If it is an Artificial Intelligence, then how it may happen
that elementary arithmetic calculations are incorrect ?

I do not believe to my eyes . . .



RELATED QUESTIONS

Errors in a billing process often lead to customer dissatisfaction and ultimately hurt... (answered by solver91311)
Bayside Insurance offers two plans. Under plan A, Gissell would pay the first $50 of the (answered by mananth)
I'm so sorry I did not give all the info on a previous question so here it is again. A (answered by lynnlo)
Among 200 patients at a Tikur Anbessa hospital. 40 of them are cardiac patients, 100 of... (answered by ikleyn)
Among 200 patients at a Tikur Anbessa hospital. 40 of them are cardiac patients, 100 of... (answered by ikleyn)
Among 200 patients at a Tikur Anbessa hospital. 40 of them are cardiac patients, 100 of... (answered by greenestamps,ikleyn)
Among 200 patients at TAH, 40 of them are cardiac patients, 100 of them are cancer... (answered by ewatrrr,ikleyn)
If x represents the amount of medical bills ($), write a function representing the amount (answered by macston)
Among 200 patients at a Tikur Anbessa hospital. 40 of them are cardiac patients, 100 of... (answered by ikleyn)