Three dice are rolled. If the probability of getting different numbers on the three dice is \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime, then \(q - p\) is equal to:
To solve this problem, we need to find the probability of getting different numbers on three dice when they are rolled. Let's calculate this step-by-step.
Total outcomes when three dice are rolled: There are 6 faces on each die. Hence, the total number of outcomes when three dice are rolled is \(6 \times 6 \times 6 = 216\).
Favorable outcomes (all different numbers):
The first die can show any of the 6 numbers.
The second die can show any number except the one that appeared on the first die. So, it has 5 options.
The third die can show any number except those that appeared on the first and second dice. So, it has 4 options.
Probability calculation: The probability of all different numbers appearing is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\(\frac{120}{216} = \frac{10}{18} = \frac{5}{9}\) (after simplifying the fraction).
Given that the probability can be expressed as \(\frac{p}{q}\) where \(p\) and \(q\) are co-prime, compare with the simplified probability \(\frac{5}{9}\), hence \(p = 5\) and \(q = 9\).
We need to find \(q - p\), which is:
\(q - p = 9 - 5 = 4\).
Therefore, the value of \(q - p\) is 4. Thus, the correct answer is 4.
