Question

A group of ASU Sustainability students took 200 water samples from groundwater in different places in the Phoenix valley. They test those samples for a certain chemical. The test is not 100% correct. If the sample contains the chemical, the test shows a positive test results 97% of the time. If the sample does not contain the chemical, it give a negative result 89% of the time. If 15 groundwater samples contain chemicals, what is the probability the sample contains a chemical if you have a positive test return for the sample?

Answer 1

Answer: 0.4169

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=
`15*0.097=14.55`

No. of sample does not contain chemical = 200-15=185

Out of 185 , number of samples does not contains chemical shows positive results =
`185*0.11=20.35`

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 :-

`\text{P(Positive }|\text{ Contains chemical})=\frac{\text{P(Positive and chemical)}}{\text{P(Positive)}}`

`=((14.55)/(200))/((34.9)/(200))=(14.55)/(34.9)\\\\=0.416905444126\approx0.4169`

Hence, the probability the sample contains a chemical if you have a positive test return for the sample =0.4169