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Bayes Theorem "AN" from ANANTPUR: 10/17 | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question
A letter is known to have arrived either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters "AN" are visible. The probability that the letter came from ANANTPUR is
A710\dfrac{7}{10}
B1017\dfrac{10}{17}correct
C1219\dfrac{12}{19}
D719\dfrac{7}{19}
Solution
Step 1: A word of length LL has L−1L-1 consecutive pairs. KANPUR: KA, AN, NP, PU, UR — "AN" once ⇒P(AN∣KANPUR)=15\Rightarrow P(\text{AN}\mid\text{KANPUR})=\dfrac15. ANANTPUR: AN, NA, AN, NT, TP, PU, UR — "AN" twice ⇒P(AN∣ANANTPUR)=27\Rightarrow P(\text{AN}\mid\text{ANANTPUR})=\dfrac27. Step 2: By Bayes with P(KANPUR)=P(ANANTPUR)=12P(\text{KANPUR})=P(\text{ANANTPUR})=\dfrac12:
P(ANANTPUR∣AN)=12⋅2712⋅15+12⋅27=2715+27.P(\text{ANANTPUR}\mid\text{AN})=\frac{\frac12\cdot\frac27}{\frac12\cdot\frac15+\frac12\cdot\frac27}=\frac{\frac27}{\frac15+\frac27}.
 Step 3: 15+27=7+1035=1735\dfrac15+\dfrac27=\dfrac{7+10}{35}=\dfrac{17}{35}:
=271735=27⋅3517=2⋅517=1017.=\frac{\frac27}{\frac{17}{35}}=\frac27\cdot\frac{35}{17}=\frac{2\cdot5}{17}=\frac{10}{17}.
 Correct answer: (2)
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