# Math Help - Conditional Probablity Problem

1. ## Conditional Probablity Problem

I was wondering if someone could help me with this...

A test for the presence of E. coli in water detects the bacteria 97% of the time when the bacteria is present, but also gives a false positive test 2% of the time, wrongly indicating the presence of E. coli in uninfected water.
If 10% of the water samples tested contain E. coli, what is the probability that a test result indicating the presence of the bacteria is accurate?

Thanks for any help!

2. Originally Posted by ty2391
I was wondering if someone could help me with this...

A test for the presence of E. coli in water detects the bacteria 97% of the time when the bacteria is present, but also gives a false positive test 2% of the time, wrongly indicating the presence of E. coli in uninfected water.
If 10% of the water samples tested contain E. coli, what is the probability that a test result indicating the presence of the bacteria is accurate?

Thanks for any help!
Draw a tree diagram!

Pr(E) = 0.1 and Pr(E') = 0.9 are the first two branches.

From E the two branches are Pr(+ve) = 0.97 and Pr(-ve) = 0.03. (Note that these probabilities are actually Pr(+ve | E) = 0.97 and Pr(-ve | E) = 0.03).

From E' the two branches are Pr(+ve) = 0.02 and Pr(-ve) = 0.98. (Note that these probabilities are actually Pr(+ve | E') = 0.02 and Pr(-ve | E') = 0.98).

Then Pr(E | +ve) = $\frac{(0.1)(0.97)}{(0.1)(0.97) + (0.9)(0.02)} = \, ....$