Download Baye`s Theorem - PatrickDooleyDHS

Baye’s Theorem
Working with Conditional
Probabilities
Drug Testing
• Suppose it is assumed that about 5% of the
population uses drugs. A drug test that is given to
employees of a certain company is 95%
accurate; that is, it correctly labels a drug user
95% of the time and correctly labels a nonuser
95% of the time. If a random employee takes the
drug test and tests positive, what is the
probability that he/she is a drug user?
Baye’s Theorem
1) P(A and B) = P(A) * P(B | A)
2) P(B and A) = P(B) * P(A | B)
Since the P(A and B) is the same as the P(B and A),
we can equate statements 1 and 2
3) P(A) * P(B | A) = P(B) * P(A | B)
Divide both sides by P(A) to get
P( B)  P( A | B)
4) P( B | A) 
P( A)
Solution to Drug Problem
Let A = event that a person tests positive on the drug test
Let B = event that a person is a drug user
We are looking for the probability that a person
is a drug user given that their test turned out positive.
P(drug user | positive test result) =
P(drug user )  P(positive test result | drug user )
P(positive test result )
P(drug user) = 0.05
P(positive test result | drug user) = 0.95
P(positive test result) =
P(positive test result and drug user) +
0.95*0.05
P(positive test result and nonuser)
0.05*0.95
given in problem
given in problem
= 0.095
0.05  0.95
 0.5
P(drug user | positive test result) =
0.05  0.95  0.95  0.05
Another Method
P(drug user | Test positive) =
475
 0.5
950
Test Positive
result
Negative
Drug Status
Drug
User
Non User
0.95*500
= 475
475
950
25
0.95*9500 9050
= 9025
500
9500
10000
Positive
Test
0.95
Drug user
0.05
0.95
Non-Drug
user
0.05*0.095 = 0.0475
0.05
Negative
Test
Positive
Test
Negative
Test
0.05*0.05 = 0.0025
0.95*0.05 = 0.0475
0.05
0.95
0.95*0.95 = 0.9025