problem 4 and 5 of bayes theorem

problem 4

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let ( 3/4 ) be the probability that he knows the answer and ( 1/4) be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability ( 1/4 ). What is the probability that the student knows the answer given that he answered it correctly?


Let E1 be the event that the  student knows the answer.

let E2 be the event that the student guesses the answer.

let A be the event that the answer is correct.

P( E1 ) = ( 3 / 4 )

P( E2 ) = ( 1 / 4 )

P( A / E1 ) = 1 [ E1  means student knows the answer and hence the answer will be correct ]

P( A / E2 ) = ( 1 / 4 )  [ given in the question  ]

Required probability = P [ student knows the answer given that he answered it correctly ]

Required probability = P [ E1 / A ]


 


P ( E1 / A ) = [ ( 3 / 4 )( 1 )] / { [ ( 3 / 4 )( 1 ) ] + [ ( 1 / 4 )( 1 / 4 ) ] }


P ( E1 / A ) =  [3] / { 3 + ( 1/4 )} =  ( 12 / 13 )

index of more problems on baye's theorem for ncert cbse mathematics


problem 5

A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?


Let E1 be the event that the person has the disease .

let E2 be the event that the person does not have the disease.

let A be the event that the test result of the person is positive.

P( E1 ) = ( 0.1 / 100 ) [0.1 percent of the population actually has the disease]

P( E2 ) = ( 99.9 / 100 ) [ complement, 100 - 0.1 = 99.9 % of the population does not have the disease ]

P( A / E1 ) = ( 99 / 100 ) [ test is 99% effective in detecting a certain disease when it is in fact, present ]

P( A / E2 ) = ( 0.5 / 100 )  [ the test also yields a false positive result for 0.5% of the healthy person tested ]

Required probability = P [ a person has the disease given that his test result is positive ]

Required probability = P [ E1 / A ]





P ( E1 / A ) = [ ( 0.1 / 100 )( 99 / 100 )] / { [( 0.1 / 100 )( 99 / 100 )] + [ ( 99.9 / 100 )( 0.5 / 100 ) ] }


P ( E1 / A ) =  [ 9.9 ] / { 9.9 + 49.95 } =  ( 9.9 / 59.85 ) = ( 990 / 5985) = ( 22 / 133 )


index of more problems on baye's theorem for ncert cbse mathematics

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