Question

A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is _________ (round off to three decimal places).

Hint:

This is a binomial probability problem where we calculate the probability of exactly two successes (sum of 4) in three trials (rolls of the dice).

Answer 1

Correct Answer: 0.018

The probability of obtaining a sum of 4 on a single roll of two six-faced dice is calculated by considering all possible outcomes that result in a sum of 4. These outcomes are: \[ (1, 3), (2, 2), (3, 1) \] So, there are 3 favorable outcomes for a sum of 4. Since there are 36 total outcomes when rolling two dice, the probability of getting a sum of 4 in one roll is: \[ P(\text{sum of 4}) = \frac{3}{36} = \frac{1}{12} \approx 0.0833 \] We are asked to find the probability that the sum of 4 occurs in exactly two of the three rolls. This is a binomial probability problem, where we want exactly two successes (sum of 4) in three trials (rolls). The binomial probability is given by: \[ P(X = 2) = \binom{3}{2} \left( \frac{1}{12} \right)^2 \left( 1 - \frac{1}{12} \right)^{3 - 2} \] \[ P(X = 2) = 3 \times \left( \frac{1}{12} \right)^2 \times \left( \frac{11}{12} \right) \] \[ P(X = 2) \approx 3 \times \frac{1}{144} \times \frac{11}{12} \approx 0.0206 \] Thus, the probability that the sum of 4 occurs in exactly two of the three rolls is: \[ \boxed{0.021} \]