Multiple choice

A six faced die is so biased that it is twice as likely to show an odd number as an even number when rolled. The probability that the sum of the numbers on the upturned faces is even when the die is thrown twice is

  1. ${\mkern 1mu} \dfrac{5}{9}$
  2. ${\mkern 1mu} \dfrac{4}{9}$
  3. ${\mkern 1mu} \dfrac{1}{2}$
  4. ${\mkern 1mu} \dfrac{1}{3}$
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A Correct answer
Explanation

Let P(odd) = 2p and P(even) = p. Since 3p + 3(2p) = 1, 9p = 1, so p = 1/9. P(even) = 1/3, P(odd) = 2/3. Sum is even if both are even or both are odd. P(even, even) = (1/3)^2 = 1/9. P(odd, odd) = (2/3)^2 = 4/9. Total probability = 1/9 + 4/9 = 5/9.

AI explanation

Let the probability of an even number be x; then the probability of an odd number is 2x. Since probabilities sum to 1, we have x + 2x = 1, meaning the probability of an even number is 1/3 and an odd number is 2/3. The sum of two die rolls is even if both are even or both are odd, so using the addition rule for mutually exclusive events, we add (1/3 * 1/3) and (2/3 * 2/3). The total probability is 1/9 + 4/9 = 5/9.

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