Respuesta :
Answer:
[tex] P(A|+) = \frac{980}{2960}=\frac{49}{148}[/tex]
Step-by-step explanation:
For this case we cna define the following notation:
+ : represent that the test is positive
-: represent that the test is negative
A= the dog present the allergy
NA= the dog not present the allergy
We know that for this case [tex] P(A) = 0.01[/tex] so then by complement [tex] P(NA)= 1-P(A) = 1-0.01= 0.99[/tex]
The accuracy on this case is 0.98 so then we have the following probability:
[tex] P(+|A) = 0.98[/tex]
Who means that if the dog present the allergy the test detect this with a 98%.
[tex] P(-|NA) = 0.98[/tex]
We can complet the following table with a basis of 10000 people:
Test + Test - Total
Allergy (A) 980 20 1000
Not Allergy (NA) 1980 97020 99000
Total 2960 97040 100000
And we want to find this probability:
[tex] P( A|+)[/tex]
And we can use the bayes rule for this case:
[tex]P( A|+) = \frac{P(A and +)}{P(+)}[/tex]
For this case we know that the probability of positive is [tex] P(+) = \frac{2960}{100000}[/tex] and the probability of [tex] P(A and +) =\frac{980}{100000}[/tex], so then we have:
[tex] P(A|+) = \frac{980}{2960}=\frac{49}{148}[/tex]
The probability that one has a dog allergy when tested positive will be 0.3311.
How to compute the probability?
From the information, 1% of the population has an allergy to dogs. This will be:
= 1% × 100000
= 1000
Therefore, 1000 people have an allergy to dogs.
The number of people who doesn't have allergy to dogs will be:
= 100000 - 1000
= 99000
Those that have allergies = 2% × 1000 = 20
Therefore, the probability that one has dog allergy will be:
= P(has allergy and test positive)
= (0.01 × 0.98) + (0.098 + 0.0198)
= 0.0098/0.0296
= 0.3311
In conclusion, the probability is 0.3311.
Learn more about probability on:
https://brainly.com/question/24756209
