Answer: 0.4169
Step-by-step explanation:
Given : If the sample contains the chemical, the probability that test shows a positive test results = 0.97
If the sample does not contain the chemical, then the probability that it give a negative result =0.89
Then, If the sample does not contain the chemical, the probability that test shows a positive test results =1-0.89=0.11
Out of 15 , number of samples contains chemical shows positive results= [tex]15\times0.097=14.55[/tex]
No. of sample does not contain chemical = 200-15=185
Out of 185 , number of samples does not contains chemical shows positive results = [tex]185\times0.11=20.35[/tex]
Sample have positive test = 14.55+20.35= 34.90
Now, the probability the sample contains a chemical if you have a positive test return for the sample :-
[tex]\text{P(Positive }|\text{ Contains chemical})=\dfrac{\text{P(Positive and chemical)}}{\text{P(Positive)}}[/tex]
[tex]=\dfrac{\dfrac{14.55}{200}}{\dfrac{34.9}{200}}=\dfrac{14.55}{34.9}\\\\=0.416905444126\approx0.4169[/tex]
Hence, the probability the sample contains a chemical if you have a positive test return for the sample =0.4169
