Multiple choice

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

  1. $\dfrac { 7 }{ 4 } $
  2. $\dfrac { 4 }{ 7 } $
  3. $\dfrac { 5 }{ 7 } $

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Probabilities are proportional to i, so P(i) = k*i. Sum of probabilities = k(1+2+3+4+5+6) = 21k = 1, so k = 1/21. Even numbers are 2, 4, 6. P(even) = P(2) + P(4) + P(6) = k(2+4+6) = 12k = 12/21 = 4/7.

AI explanation

Using the properties of proportionality, let the probabilities of faces 1 through 6 be k, 2k, 3k, 4k, 5k, and 6k. Since the sum of all probabilities must equal 1, we have 21k = 1, which means k = 1/21. The probability of rolling an even number is the sum of the probabilities for faces 2, 4, and 6, which is 2k + 4k + 6k = 12k. Substituting k = 1/21 gives 12/21, which simplifies to 4/7. The result is 4/7.

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