A box contains two dice: a fair die, and a die whose six faces show the numbers 2, 2, 4, 4, 6, 6. One die is chosen at random from the box and thrown. Consider the following statements:
(A) The probability that the number shown is even is \(\displaystyle\frac{3}{4}\).
(B) If the number shown is even, the probability that the fair die was thrown is \(\displaystyle\frac{1}{3}\).
(C) The probability that the number shown is 6 is \(\displaystyle\frac{1}{4}\).
(D) If the number shown is odd, the probability that the fair die was thrown is \(\displaystyle\frac{1}{2}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
Let \(E_1\) and \(E_2\) be the events that the fair die and the other die are chosen. Each has probability \(\frac{1}{2}\).
The fair die shows an even number with probability \(\frac{1}{2}\) and a 6 with probability \(\frac{1}{6}\). The other die always shows an even number, and shows 6 on 2 of its 6 faces.
By the theorem of total probability:
\( \displaystyle P(\text{even}) = \frac{1}{2}\times\frac{1}{2}+\frac{1}{2}\times 1 = \frac{3}{4} \)
\( \displaystyle P(6) = \frac{1}{2}\times\frac{1}{6}+\frac{1}{2}\times\frac{2}{6} = \frac{1}{12}+\frac{2}{12} = \frac{1}{4} \)
By Bayes’ theorem:
\( \displaystyle P(E_1|\text{even}) = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \)
An odd number can come only from the fair die, so \(P(E_1|\text{odd}) = 1\).
So P(even) = \(\frac{3}{4}\), P(fair die | even) = \(\frac{1}{3}\), P(6) = \(\frac{1}{4}\) and P(fair die | odd) = 1.
Checking the statements
(A) The probability that the number shown is even is \(\frac{3}{4}\). — Correct. \(\frac{1}{4}+\frac{1}{2} = \frac{3}{4}\).
(B) If the number shown is even, the probability that the fair die was thrown is \(\frac{1}{3}\). — Correct. \(\frac{1}{4}\div\frac{3}{4} = \frac{1}{3}\).
(C) The probability that the number shown is 6 is \(\frac{1}{4}\). — Correct. \(\frac{1}{12}+\frac{2}{12} = \frac{1}{4}\).
(D) If the number shown is odd, the probability that the fair die was thrown is \(\frac{1}{2}\). — Incorrect. The other die has no odd face, so this probability is 1.