In the last week due to coronavirus outbreak, 200 children were tested positive, 400 adults were tested positive and 600 senior citizens were tested positive.The probability of them dying are 0.01, 0.03 and 0.15 respectively. If one of the person tested positive dies due to coronavirus what is the probability that he/she was a senior citizen.

Concept:

Bayes' Theorem:

Let E1, E2, ….., En be n mutually exclusive and exhaustive events associated with a random experiment and let S be the sample space. Let A be any event which occurs together with any one of E1 or E2 or … or En such that P(A) ≠ 0. Then

\(P\left( {{E_i}\;|\;A} \right) = \frac{{P\left( {{E_i}} \right)\; \times \;P\left( {A\;|\;{E_i}} \right)}}{{\mathop \sum \nolimits_{i\; = 1}^n P\left( {{E_i}} \right) \times P\left( {A\;|\;{E_i}} \right)}},\;i = 1,\;2,\; \ldots .\;,\;n\)

Calculation:

Let the events B1, B2 and B3 be defined as:

B1 : Children who were found coronavirus positive in the last week

B2 : Adults who were found coronavirus positive in the last week

B3 : Senior citizens who were found coronavirus positive in the last week

Let the event E be 'The person tested coronavirus positive died'

⇒P (B₁) = 200/1200 = 1/6, P  (B₂) = 400/1200 = 1/3 and P (B₃) = 600/1200 = 1/2

Similarly, P (E | B1) = 0.01, P (E | B2) = 0.03 and P (E | B3) = 0.15 ------(Given)

Here, we have to find the value of P (B₃ | E)

As we know that according to bayes' theorem: \(P\left( {{E_i}\;|\;A} \right) = \frac{{P\left( {{E_i}} \right)\; \times \;P\left( {A\;|\;{E_i}} \right)}}{{\mathop \sum \nolimits_{i\; = 1}^n P\left( {{E_i}} \right) \times P\left( {A\;|\;{E_i}} \right)}},\;i = 1,\;2,\; \ldots .\;,\;n\)

\( \Rightarrow P{\rm{(}}{B_3}\;{\rm{|}}E) = \frac{{P\left( {{B_3}} \right) \cdot P\;\left( {E\;|\;{B_3}} \right)}}{{\left[ {P\left( {{B_1}} \right) \cdot P\;\left( {E\;|\;{B_1}} \right) + \;P\left( {{B_2}} \right) \cdot P\;\left( {E\;|\;{B_2}} \right) + P\left( {{B_3}} \right) \cdot P\;\left( {E\;|\;{B_3}} \right)} \right]}}\;\)

\( \Rightarrow P{\rm{(}}{B_3}\;{\rm{|}}E) = \frac{45}{{52}}\)