in a certain board game, a 12 - sided number cube showing numbers 1 - 12 is rolled. if 4 such number cubes…

Explanation:

Step1: Find probability of a single - cube showing less than 8

The numbers less than 8 on a 12 - sided cube are 1, 2, 3, 4, 5, 6, 7. So the probability of a single 12 - sided cube showing a number less than 8 is $p=\frac{7}{12}$.

Step2: Find probability of all 4 cubes showing less than 8

Since the rolls are independent events, the probability that all 4 cubes show a number less than 8 is $P(\text{all}<8)=(\frac{7}{12})^4$. Calculate $(\frac{7}{12})^4=\frac{7^4}{12^4}=\frac{2401}{20736}\approx0.1158$.

Step3: Find probability of at least 1 cube showing 8 or larger

The probability of at least 1 cube showing 8 or larger is the complement of the event that all cubes show less than 8. Let $P(X\geq1)$ be the probability that at least 1 cube shows 8 or larger. Then $P(X\geq1)=1 - P(\text{all}<8)$. $P(X\geq1)=1-\frac{2401}{20736}=\frac{20736 - 2401}{20736}=\frac{18335}{20736}\approx0.8842$.

Answer:

0.8842