Probability Bayes formula Question

[crays]

Hi guys, I've got 2 probability questions that i couldn't solve. Please help.

1) A computer program has an error that causes the program not to function perfectly. n programmer are assigned separately to detect the error. The probability that each programmer will detect the error is 0.875. Determine the value of n if the probability that at least one programmer detects the error is 0.998


2) A new skin test is devised for detecting tuberculosis. To evaluate the test before it is put into use, a research indicates tuberculosis in 96% of those who have it and in 2% of those who do not. It is known that 8% of the population has tuberculosis.

a) Find the probability of a randomly selected person having tuberculosis given that the skin test is positive.

I've gotten this right, Its (Have tuberculosis|Test is positive) so what i did was using (0.96 x 0.08) + (0.02 x 0.92) the value for all test is positive. Then 0.96 x 0.08 divide by the value i just got, i got the answer right.

b) Find the probability that a person has tuberculosis given that the test indicates no tuberculosis is present.

What i had in mind is to reverse the probability for all test is positive and get all test is negative and do the same, but it wasn't right.

c) Find the probability of the skin test giving a false positive result.

Please help :) Thanks

 

[LCKurtz]

You would do well to use symbols and equations. For part (a) if you let X = number of programmers finding the error then X is binomial with parameters n and p = .875. You are looking for P(X ≥ 1) which is 1 - P(X = 0) to be .998.

For the second part, again, use some mathematical notation.

Let P be the event the test is positive
Let T be the event the patient has tuberculosis.

In part (a) you were calculating P(T|P) using the (unwritten) Bayes formula. To make your solution readable you should write:

$$P(T|P) = \frac {P(P|T)P(T)}{P(P|T)P(T) + P(P|\overline{T})P(\overline{T})} =$$

and put your numbers in after the = sign.

For (b) you are asking for $$P(T|\overline{P})$$. If you will begin by writing out the formula for that you may see how to calculate it. Using the formulas can make all the difference.

 

[Architeuthis Dux]

For the first question, we can use the Bayes formula to solve for n:

P(at least one programmer detects error) = 1 - P(no programmer detects error)

= 1 - (0.125)^n

= 0.998

Solving for n, we get n = 5.

For the second question, we can use the Bayes formula again to solve for the probabilities:

a) P(have tuberculosis | test is positive) = P(test is positive | have tuberculosis) * P(have tuberculosis) / P(test is positive)

= (0.96 * 0.08) / (0.96 * 0.08 + 0.02 * 0.92)

= 0.96

b) P(have tuberculosis | test is negative) = P(test is negative | have tuberculosis) * P(have tuberculosis) / P(test is negative)

= (0.04 * 0.08) / (0.04 * 0.08 + 0.98 * 0.92)

= 0.0004

c) P(false positive) = P(test is positive | no tuberculosis) * P(no tuberculosis)

= 0.02 * 0.92

= 0.0184

Hope this helps!