The tree diagram shows two levels of branches for all of the outcomes. The first

Question

The tree diagram shows two levels of branches for all of the outcomes. The first level has a header denoting pet allergy status outcomes and starts as two branches, with the top branch identified as pet allergy, with probability of 0.18, and the bottom branch identified as no pet allergy, with probability of 0.82. The second level of branches display two branches for each of the pet allergy test outcomes. The branches stemming from the pet allergy status show a top branch for a test result of “positive” with conditional probability of 0.81, and a bottom branch for a test result of “negative” with conditional probability of 0.19. The branches stemming from the no pet allergy status show a top branch for a test result of “positive” with conditional probability of 0.06, and a bottom branch for a test result of “negative” with conditional probability of 0.94. Suppose a county’s recent health report gives a pet allergy prevalence of 0.18 for kids. There is a new at-home test kit for pet allergies on the market that provides families with a convenient way to test if children have a pet allergy. The probability that the test gives a positive result when a child really does not have a pet allergy is 0.06. The probability that the test gives a negative result when a child really does have a pet allergy is 0.19. The tree diagram shows the possible outcomes with the associated probabilities. What is the probability of an incorrect test result? Give your answer as a decimal precise to two decimal places.

Explanation / Answer

According to the question, we have to find the probability of an incorrect test result.

for that purpose, we need to find i) the probability that the test gives positive pet allergy result and the person don't have pet allergy.

and also, ii) the probability that the test gives negative pet allergy result and the person has the pet allergy.

now, consider the events,

A: the person has pet allergy

B: the test shows positive results.

now, clearly, by the tree diagram, we get,

P(A)=0.18 , P(Ac)=0.82

by the given condition that,The probability that the test gives a positive result when a child really does not have a pet allergy is 0.06. The probability that the test gives a negative result when a child really does have a pet allergy is 0.19.

we get,

P(B|Ac)=0.6 , P(Bc|A)=0.19

now, according to our problem statements,we need to find the probabilities,

i) P(B*Ac) , ii)P(Bc*A)

for i) P(B*Ac) = P(B|Ac)*P(Ac) , from the formula of conditional probability

= 0.6*0.82 = 0.492

for ii) P(Bc*A) = P(Bc|A)*P(A) , from the formula of conditional probability

= 0.19*0.18 = 0.0342

Hence, the probability of an incorrect test result = P(B*Ac) + P(Bc*A) = 0.492 + 0.0342 = 0.5262 = 0.53 (approx)

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