Multiple choice

A six- faced dice is so biased that it is twice as likely to show an even number as an odd number when thrown. it is thrown twice. The probability that the sum of two numbers thrown is even is :

  1. $1/12$
  2. $1/6$
  3. $1/3$
  4. $5/9$
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D Correct answer
Explanation

The probability of an even result is twice the probability of an odd result, so P(even) = 2/3 and P(odd) = 1/3. An even sum occurs when both results have the same parity, giving (2/3)^2 + (1/3)^2 = 5/9.

AI explanation

Let the probability of an odd number be p, making the probability of an even number 2p, so p plus 2p equals 1, which means p is 1/3 and the probability of an even number is 2/3. The sum of two numbers will be even if both numbers are even or both are odd, calculated using the sum of independent probabilities. The probability is (2/3 times 2/3) plus (1/3 times 1/3), which is 4/9 plus 1/9, equaling 5/9. The required probability is 5/9.

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