Question 2 - Case Based Questions (MCQ) - Chapter 13 Class 12 Probability
Last updated at May 29, 2023 by Teachoo
The reliability of a COVID PCR test is specified as follows:
Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/her as COVID positive.
Based on the above information, answer the following:
Question 1
What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually having COVID?
(a) 0.001
(b) 0.1
(c) 0.8
(d) 0.9
Question 2
What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually not having COVID’?
(a) 0.01
(b) 0.99
(c) 0.1
(d) 0.001
Question 3
What is the probability that the ‘person is actually not having COVID?
(a) 0.998
(b) 0.999
(c) 0.001
(d) 0.111
Question 4
What is the probability that the ‘person is actually having COVID given that ‘he is tested as COVID positive’?
(a) 0.83
(b) 0.0803
(c) 0.083
(d) 0.089
Question 5
What is the probability that the ‘person selected will be diagnosed as COVID positive’?
(a) 0.1089
(b) 0.01089
(c) 0.0189
(d) 0.189
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Question The reliability of a COVID PCR test is specified as follows: Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/her as COVID positive. Based on the above information, answer the following:Given,
Of people having COVID, 90% of the test detects the disease but 10% goes undetected.
Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive.
90% detected
10% undetected
Does not have COVID
99% COVID negative
1% COVID positive
Let,
E : The event that person selected has COVID
F : The event that person selected does not have COVID
G : The event that person is tested positive
Also,
From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/her as COVID positive.
P(person has COVID) = P(E) = 0.1%
= 0.1 Γ 1/100 = 0.001
Question 1 What is the probability of the βperson to be tested as COVID positiveβ given that βhe is actually having COVID? (a) 0.001 (b) 0.1 (c) 0.8 (d) 0.9Has COVID
90% detected
10% undetected
P(tested COVID | has COVID)
= P(G|E)
= 90%
= 90/100 = 0.9
So, the correct answer is (d)
Question 2 What is the probability of the βperson to be tested as COVID positiveβ given that βhe is actually not having COVIDβ? (a) 0.01 (b) 0.99 (c) 0.1 (d) 0.001 Does not have COVID
99% COVID negative
1% COVID positive
P(tested COVID positive| does not has COVID) = P(G|F)
= 1%
= 1/100
= 0.01
So, the correct answer is (a)
Question 3 What is the probability that the βperson is actually not having COVID? (a) 0.998 (b) 0.999 (c) 0.001 (d) 0.111 P(person not having COVID) = 1 β P(person having COVID)
P(F) = 1 β P(E)
= 1β 0.001
= 0.999
So, the correct answer is (b)
Question 4 What is the probability that the βperson is actually having COVID given that βhe is tested as COVID positiveβ? (a) 0.83 (b) 0.0803 (c) 0.083 (d) 0.089
We need to find
Probability that the person is actually having COVID given that he is tested as COVID positive
i.e. P (having covid | covid positive)
i.e. P (E|G)
Now,
P(E|G) = (π(πΈ). π(πΊ|πΈ))/(π(πΈ). π(πΊ|πΈ)+π(πΉ). π(πΊ|πΉ))
"P(E)"
i.e. Probability that the person has COVID
P(E) = 0.1%=π.πππ
P(G|E)
i.e. P(tested COVID positive | has COVID)
This is calculated in Question 1
π("G|E") = 0.9
"P(F)"
i.e. Probability that the person does not have COVID
This is calculated in Question 3
"P(F)" = 0.999
P(G|F)
i.e. P(tested COVID positive| does not has COVID)
This is calculated in Question 2
π("G|" πΉ) = 0.01
Putting values in formula,
P(E|G) = (0.001 Γ 0.9)/(0.001 Γ 0.9 + 0.999 Γ 0.01)
= 0.0009/(0.0009 + 0.0099)
= (π.ππππ)/(π.πππππ)
= (9/1000)/(1089/100000)
= (9 Γ 10)/1089
= 0.083 ( approx.)
So, the correct answer is (c)
Question 5 What is the probability that the βperson selected will be diagnosed as COVID positiveβ? (a) 0.1089 (b) 0.01089 (c) 0.0189 (d) 0.189P(person selected will be diagnosed as COVID positive)
= P (person does not have covid and is covid positive)
+ P(person has covid and is covid positive)
= P(does not have covid) Γ P(tested COVID positive | does not have COVID)
+ P(has covid) Γ P(tested COVID positive | has COVID)
= P(F). P(G|F) + P(E). P(G|E)
= 0.001 Γ 0.9 + 0.999 Γ 0.01
= 0.0009 + 0.0099
= 0.01089
So, the correct answer is (b)
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.
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