GATE 2021 EC – Question 37
A box contains the following three coins.
I. A fair coin with head on one face and tail on the other face.
II. A coin with heads on both the faces.
III. A coin with tails on both the faces.
A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is
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Show answer and explanation
Correct answer: (B) $\frac13$
Explanation
$P(\text{first head})=\frac13\cdot\frac12+\frac13\cdot1+0=\frac12$. If the first coin was I (joint probability $\frac16$ with a head), the remaining coins are II and III and the second toss is a head with probability $\frac12$. If the first coin was II (probability $\frac13$), the remaining coins are I and III and the second toss is a head with probability $\frac12\cdot\frac12=\frac14$. So $P(\text{both heads})=\frac16\cdot\frac12+\frac13\cdot\frac14=\frac16$ and $P=\frac{1/6}{1/2}=\frac13$.