A and B each throw a fair die once. If it is known that the number thrown by A is greater than the number thrown by B, then the probability that A has thrown a 6 is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{3}\)
The 36 pairs (A's number, B's number) are equally likely. If A throws a, B must throw one of the a − 1 smaller numbers, so the pairs with A's number greater are
\(0+1+2+3+4+5 = 15\)
This is the reduced sample space. A throws a 6 in 5 of these pairs (B throws 1, 2, 3, 4 or 5). So
\( \displaystyle P = \frac{5}{15} = \frac{1}{3} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{5}{12}\) — Incorrect. This is \(\frac{15}{36}\), the probability that A's number is greater, which is the condition itself.
Option (3) \(\displaystyle\frac{1}{6}\) — Incorrect. This is the chance of A throwing a 6 when nothing is known about the throws.
Option (4) \(\displaystyle\frac{5}{36}\) — Incorrect. This is the probability that A throws a 6 and A's number is greater, not divided by \(\frac{15}{36}\).