in a certain board game, a 12 - sided number cube showing numbers 1 - 12 is rolled. if three such number…

in a certain board game, a 12 - sided number cube showing numbers 1 - 12 is rolled. if three such number cubes are rolled, what is the probability that all three show a number 10 or larger?\n$\\left(\\frac{1}{12}\\right)^3$\n$\\left(\\frac{2}{12}\\right)^3$\n$\\left(\\frac{3}{12}\\right)^3$\n$\\left(\\frac{10}{12}\\right)^3$

in a certain board game, a 12 - sided number cube showing numbers 1 - 12 is rolled. if three such number cubes are rolled, what is the probability that all three show a number 10 or larger?\n$\\left(\\frac{1}{12}\\right)^3$\n$\\left(\\frac{2}{12}\\right)^3$\n$\\left(\\frac{3}{12}\\right)^3$\n$\\left(\\frac{10}{12}\\right)^3$

Answer

Explanation:

Step1: Find single - cube probability

The numbers 10, 11, 12 are 10 or larger on a 1 - 12 numbered 12 - sided cube. So there are 3 favorable outcomes out of 12. The probability of getting a number 10 or larger on one roll is $\frac{3}{12}$.

Step2: Use multiplication rule for independent events

Since the rolls of the three number cubes are independent events, the probability that all three show a number 10 or larger is the product of the probabilities of each individual roll. So the probability is $\left(\frac{3}{12}\right)\times\left(\frac{3}{12}\right)\times\left(\frac{3}{12}\right)=\left(\frac{3}{12}\right)^3$.

Answer:

$\left(\frac{3}{12}\right)^3$