If 5.9% of patients have the flu, then 94.1% do not
When a patient does not have the flu, the nurse can correctly confirm 94% of the time; nurse is wrong 6% of the time.
When a patient does have the flu, the nurse can correctly confirm 98% of the time; nurse is wrong 2% of the time.
So this gives us the following probabilities.
1) has the flu and the nurse confirms = .059 * .98
2) has the flu and nurse does not confirm = .059 * .02
3) does not have the flu and nurse agrees = .941 * .94
4) does not have the flue but nurse says patient does have the flu = .941 * .06
This gives us the following
1) .05782 has flu properly diagnosed
2) .00118 has flu but not diagnosed
3) .88454 does not have flu and properly diagnosed
4) .05646 does not have flu and not correctly diagnosed
Notice that these 4 probabilities add to 1.0000
The probability that the nurse says a patient is infected is the sum of 1) and 4)
.05782 + .05635 = .11428
The probability that a patient has the flu and the nurse confirms 1)
.05782
The probability that a patient has the flu given that the nurse confirms
.05782 / .11428 = .50595
This type of question works best with a decision tree or a two-way table. If you would contact me directly, I'd be happy to provide you with an appropriate graphic. (sorry, can't attach images here)
