Example 20 - A doctor is to visit a patient. From past experience - Bayes theoram

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Example 20 A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively 3﷮10﷯ , 1﷮5﷯ , 1﷮10﷯ and 2﷮5﷯ . The probabilities that he will be late are 1﷮4﷯ , 1﷮3﷯ and 1﷮12﷯ if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probability that he comes by train? Let T1 : Event that doctor came by Train T2 : Event that doctor came by Bus T3 : Event that doctor came by Scooter T4 : Event that doctor came by other means of transport L : Event that doctor arrives late We need to find out Probability of doctor comes by train if he is late i.e. P(T1|L) = 𝑃 𝑇1﷯ × 𝑃(𝐿|𝑇1)﷮𝑃 𝑇1﷯ × 𝑃 𝐿|𝑇1﷯ + 𝑃 𝑇1﷯ × 𝑃 𝐿|𝑇1﷯ + 𝑃 𝑇1﷯ × 𝑃 𝐿|𝑇1﷯ + 𝑃 𝑇1﷯ × 𝑃(𝐿|𝑇1)﷯ Putting values in formula, P(T1|L) = 𝑃 𝑇1﷯ × 𝑃(𝐿|𝑇1)﷮𝑃 𝑇1﷯ × 𝑃 𝐿|𝑇1﷯ + 𝑃 𝑇1﷯ × 𝑃 𝐿|𝑇1﷯ + 𝑃 𝑇1﷯ × 𝑃 𝐿|𝑇1﷯ + 𝑃 𝑇1﷯ × 𝑃(𝐿|𝑇1)﷯ = 3﷮10 ﷯ × 1﷮4﷯﷮ 3﷮10 ﷯ × 1﷮4﷯ + 1﷮5 ﷯ × 1﷮3﷯ + 1﷮10 ﷯ × 1﷮12﷯ + 2﷮5 ﷯ × 0 ﷯ = 3﷮ 40 ﷯ ﷮ 18﷮20 ﷯ ﷯ = 3﷮40﷯ × 120﷮18﷯ = 360﷮720﷯ = 𝟏﷮𝟐﷯

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