Question

A doctor is to visit a patient. From the past experience it is known that the probabilities of the doctor coming by train, bus, scooter or taxi are $1 over 10 comma space 1 fifth comma space 3 over 10 space and space 2 over 5$ respectively. The probabilities that he will be late are $1 fourth comma space 1 third space and space 1 over 12$ if he comes by train, bus or scooter respectively but by taxi he will not be late. When he arrives he is late What is the probability that he came by bus?

Solution

Let E be the event that the doctor visits the patient late and let T₁, T₂, T₃, T₄ be the event that the doctor comes by train, bus, scooter or taxi.
Then

straight P left parenthesis straight T subscript 1 right parenthesis space equals 1 over 10 comma space space straight P left parenthesis straight T subscript 2 right parenthesis space equals space 1 fifth comma space space straight P left parenthesis straight T subscript 3 right parenthesis space equals space 3 over 10 comma space space straight P left parenthesis straight T subscript 4 right parenthesis space equals space 2 over 5

straight P left parenthesis straight E space vertical line thin space straight T subscript 1 right parenthesis space equals space Probability space that space the space doctor space arriving space late space comes space by space train space equals space 1 fourth

Similarly,

straight P left parenthesis straight E space vertical line thin space straight T subscript 2 right parenthesis space equals space 1 third comma space space straight P left parenthesis straight E space vertical line thin space straight T subscript 3 right parenthesis space equals space 1 over 12 comma space space straight P left parenthesis straight E space vertical line thin space straight T subscript 4 right parenthesis space equals space 0

By Baye's Theorem,

straight P left parenthesis straight T subscript 2 space vertical line thin space straight E right parenthesis space equals space Probability space that space the space doctor space arriving space late space comes space by space bus

equals space fraction numerator straight P left parenthesis straight T subscript 2 right parenthesis thin space straight P left parenthesis straight E space vertical line space straight T subscript 2 right parenthesis over denominator straight P left parenthesis straight T subscript 1 right parenthesis space straight P left parenthesis straight E space vertical line thin space straight T subscript 1 right parenthesis space plus space straight P left parenthesis straight T subscript 2 right parenthesis thin space straight P left parenthesis straight E space vertical line thin space straight T subscript 2 right parenthesis space plus space straight P left parenthesis straight T subscript 3 right parenthesis thin space straight P left parenthesis straight E space vertical line straight T subscript 3 right parenthesis space plus space straight P left parenthesis straight T subscript 4 right parenthesis thin space straight P left parenthesis straight E space vertical line space straight T subscript 4 right parenthesis end fraction equals space fraction numerator begin display style 1 fifth end style cross times begin display style 1 third end style over denominator begin display style 1 over 10 end style cross times begin display style 1 fourth end style plus begin display style 1 fifth end style cross times begin display style 1 third end style plus begin display style 3 over 10 end style cross times begin display style 1 over 12 end style plus begin display style 2 over 5 end style cross times 0 end fraction space equals fraction numerator begin display style 1 over 15 end style over denominator begin display style 1 over 40 end style plus begin display style 1 over 15 end style plus begin display style 1 over 40 end style plus 0 end fraction equals space fraction numerator begin display style 1 over 15 end style over denominator begin display style fraction numerator 3 plus 8 plus 3 over denominator 120 end fraction end style end fraction space equals fraction numerator begin display style 1 over 15 end style over denominator begin display style 14 over 120 end style end fraction space equals 1 over 15 cross times 120 over 14 equals space 4 over 7

For Daily Free Study Material Join wiredfaculty

Probability

Some More Questions From Probability Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location