A fair coin is tossed three times. If it is known that at least one head has appeared, then the probability that all three tosses show heads is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{7}\)
Three tosses give 8 equally likely outcomes. Only TTT has no head, so the given condition leaves a reduced sample space of 7 outcomes:
HHH, HHT, HTH, THH, HTT, THT, TTH
Only HHH shows three heads:
\( \displaystyle P = \frac{1}{7} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{8}\) — Incorrect. This is the probability of three heads without the given condition.
Option (2) \(\displaystyle\frac{3}{7}\) — Incorrect. This is the probability of exactly one head (HTT, THT, TTH) given at least one head.
Option (4) \(\displaystyle\frac{7}{8}\) — Incorrect. This is the probability of at least one head, the condition itself.