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Memory Based GATE DA 2024 | Question: 14

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Consider two events $\mathrm{T}$ and $\mathrm{S}$. Let $\overline{\mathrm{T}}$ denote the complement of event $\mathrm{T}$. The probabilities associated with different events are given as follows: $\mathrm{P}({\mathrm{T}})=0.4$, $\mathrm{P}(\mathrm{S}|\mathrm{T})=0.3$, $\mathrm{P}(\mathrm{S}|\overline{\mathrm{T}})=0.6$. Find the conditional probability $\mathrm{P}(\mathrm{T}|\mathrm{S})$.
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P(T) = 0.4 => P(T') = 0.6

P(S|T) = P(S intersection T)/P(T) => 0.3 = P(S intersection T)/0.4 => P(S intersection T) = 0.12

P(S|T') = P(S intersection T')/P(T') => 0.6 = P(S intersection T')/0.6 => P(S intersection T') = 0.36

P(S) = 0.12+0.36 = 0.48

P(T|S) = P(S intersection T)/P(S) = 0.12/0.48 = 0.25
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