Multiple choice

A die is loaded so that the probability of face $i$ is proportional to $i,i=1,2,3,...,6$. The probability of an even number occurring when the die is rolled, is

  1. $\displaystyle \frac{2}{7}$
  2. $\displaystyle \frac{4}{7}$
  3. $\displaystyle \frac{3}{7}$

  4. None of these
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B Correct answer
Explanation

The probability of face i is k*i. Sum of probabilities = k(1+2+3+4+5+6) = 21k = 1, so k = 1/21. Even numbers are 2, 4, 6. Their probabilities are 2/21, 4/21, and 6/21. Sum = (2+4+6)/21 = 12/21 = 4/7.

AI explanation

Since the probability is proportional to the face value, the probabilities are k, 2k, 3k, 4k, 5k and 6k. The sum of all probabilities must equal 1, so 21k equals 1, making k equal to 1/21. The probability of rolling an even number is the sum of the probabilities for faces 2, 4 and 6, calculated as 2k plus 4k plus 6k, which equals 12/21. Simplifying the fraction 12/21 gives the final probability of 4/7.

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