Question

If 12 dice are thrown at a time, then the probability that a multiple of 3 does not appear on any die is:

• A. \( \left( \frac{1}{2} \right)^{12} \)
• B. \( \left( \frac{1}{3} \right)^{12} \)
• C. \( \left( \frac{2}{3} \right)^{12} \)
• D. \( \left( \frac{5}{6} \right)^{12} \)

Hint:

When calculating the probability of an event not happening, subtract the probability of the event happening from 1. For independent events, multiply the probabilities of each event.

Answer 1

The Correct Option is C

We are asked to find the probability that a multiple of 3 does not appear on any of the 12 dice when thrown.
Step 1: Each die has 6 faces, numbered from 1 to 6. Out of these, the multiples of 3 are 3 and 6. Thus, there are 2 faces on the die that are multiples of 3. The probability that a multiple of 3 does not appear on a single die is the probability that the die shows either 1, 2, 4, or 5. There are 4 such faces out of 6.
Thus, the probability of not getting a multiple of 3 on one die is: \[ P(\text{not multiple of 3 on a die}) = \frac{4}{6} = \frac{2}{3} \] Step 2: Since the dice are thrown independently, the probability that none of the 12 dice shows a multiple of 3 is the product of the individual probabilities for each die. Therefore, the required probability is: \[ P(\text{no multiple of 3 on any die}) = \left( \frac{2}{3} \right)^{12} \] Thus, the probability that a multiple of 3 does not appear on any die is \( \left( \frac{2}{3} \right)^{12} \).