Question

1. you roll a pair of number cubes. what is the probability of getting even numbers on both cubes?

Explanation:

Step1: Determine the total number of outcomes

When rolling a pair of number cubes (each cube has 6 faces numbered from 1 - 6), the total number of outcomes is \(n(S)=6\times6 = 36\) (by the fundamental counting principle: if one event has \(m\) possible outcomes and another independent event has \(n\) possible outcomes, then the total number of outcomes for the two - event combination is \(m\times n\)).

Step2: Determine the number of favorable outcomes

For the first cube to show an even number (2, 4, or 6), there are 3 possibilities. For the second cube to show an even number (2, 4, or 6), there are 3 possibilities. The number of favorable outcomes \(n(A)=3\times3=9\) (using the fundamental counting principle again for the two independent events of getting an even number on the first cube and an even number on the second cube).

Step3: Calculate the probability

The probability formula is \(P(A)=\frac{n(A)}{n(S)}\). Substitute \(n(A) = 9\) and \(n(S)=36\) into the formula: \(P(A)=\frac{9}{36}=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\)