Probability Question - None

Question

A dice is loaded so that the probability of a face i is proportional to i², i = 1, 2, …, 6. Then find the probability of occurring a prime number when the dice is rolled.

Accepted Answer

Step 1: Set up the probability distribution. Since P(face i) ∝ i², we write:

P(i) = k·i² where k is the constant of proportionality

Step 2: Find k using normalization condition ΣP(i) = 1:

k(1² + 2² + 3² + 4² + 5² + 6²) = 1

k(1 + 4 + 9 + 16 + 25 + 36) = 1

k(91) = 1

k = 1/91

Step 3: Identify prime numbers on the dice: {2, 3, 5}

Step 4: Calculate probability of prime:

P(prime) = P(2) + P(3) + P(5)

P(prime) = (1/91)(2² + 3² + 5²)

P(prime) = (1/91)(4 + 9 + 25)

P(prime) = 38/91

Question:

Step-by-Step Solution

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