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A letter is known to have come either from 'TATA NAGAR' or from 'CALCUTTA'. On the envelope, just two consecutive letters TA are visible. What is the probability that the letter came from 'TATA NAGAR'?
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Let $\mathrm{E}_1$ and $\mathrm{E}_2$ be the events of the letter being from TATA NAGAR and CALCUTTA respectively.
Also, let $E_3$ be the event that on the letter, two consecutive letters TA are visible.
$\therefore \quad \mathrm{P}\left(\mathrm{E}_1\right)=\frac{1}{2}$ and $\mathrm{P}\left(\mathrm{E}_2\right)=\frac{1}{2}$
and $\mathrm{P}\left(\mathrm{E}_3 / \mathrm{E}_1\right)=\frac{2}{8}$ and $\mathrm{P}\left(\mathrm{E}_3 / \mathrm{E}_2\right)=\frac{1}{7}$
$$
\mathrm{P}\left(\mathrm{E}_1 / \mathrm{E}_3\right)=\frac{\frac{1}{2} \cdot \frac{2}{8}}{\frac{1}{2} \cdot \frac{2}{8}+\frac{1}{2} \cdot \frac{1}{7}}=\frac{\frac{1}{8}}{\frac{11}{56}}=\frac{7}{11}
$$
[Baye's Theorem]
Also, let $E_3$ be the event that on the letter, two consecutive letters TA are visible.
$\therefore \quad \mathrm{P}\left(\mathrm{E}_1\right)=\frac{1}{2}$ and $\mathrm{P}\left(\mathrm{E}_2\right)=\frac{1}{2}$
and $\mathrm{P}\left(\mathrm{E}_3 / \mathrm{E}_1\right)=\frac{2}{8}$ and $\mathrm{P}\left(\mathrm{E}_3 / \mathrm{E}_2\right)=\frac{1}{7}$
$$
\mathrm{P}\left(\mathrm{E}_1 / \mathrm{E}_3\right)=\frac{\frac{1}{2} \cdot \frac{2}{8}}{\frac{1}{2} \cdot \frac{2}{8}+\frac{1}{2} \cdot \frac{1}{7}}=\frac{\frac{1}{8}}{\frac{11}{56}}=\frac{7}{11}
$$
[Baye's Theorem]
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