Question
Suppose you roll a red number cube and a blue number cube. What is the probability that you will roll a 3 on the red cube and an even number on the blue cube
Answers
GPT-4o mini
To solve this problem, we need to determine the probability of two independent events happening: rolling a 3 on the red cube and rolling an even number on the blue cube.
1. **Probability of rolling a 3 on the red cube**: A standard number cube (or die) has 6 faces numbered from 1 to 6. The probability of rolling a specific number (in this case, a 3) is:
\[
P(\text{rolling a 3 on the red cube}) = \frac{1}{6}
\]
2. **Probability of rolling an even number on the blue cube**: The even numbers on a standard number cube are 2, 4, and 6. There are 3 even numbers out of the 6 possible outcomes, so the probability is:
\[
P(\text{rolling an even number on the blue cube}) = \frac{3}{6} = \frac{1}{2}
\]
3. **Calculating the combined probability**: Since the rolls of the cubes are independent events, we multiply the probabilities of the two events:
\[
P(\text{3 on red and even on blue}) = P(\text{rolling a 3 on the red cube}) \times P(\text{rolling an even number on the blue cube}) = \frac{1}{6} \times \frac{1}{2}
\]
\[
P(\text{3 on red and even on blue}) = \frac{1}{12}
\]
Therefore, the probability of rolling a 3 on the red cube and an even number on the blue cube is \(\frac{1}{12}\).
1. **Probability of rolling a 3 on the red cube**: A standard number cube (or die) has 6 faces numbered from 1 to 6. The probability of rolling a specific number (in this case, a 3) is:
\[
P(\text{rolling a 3 on the red cube}) = \frac{1}{6}
\]
2. **Probability of rolling an even number on the blue cube**: The even numbers on a standard number cube are 2, 4, and 6. There are 3 even numbers out of the 6 possible outcomes, so the probability is:
\[
P(\text{rolling an even number on the blue cube}) = \frac{3}{6} = \frac{1}{2}
\]
3. **Calculating the combined probability**: Since the rolls of the cubes are independent events, we multiply the probabilities of the two events:
\[
P(\text{3 on red and even on blue}) = P(\text{rolling a 3 on the red cube}) \times P(\text{rolling an even number on the blue cube}) = \frac{1}{6} \times \frac{1}{2}
\]
\[
P(\text{3 on red and even on blue}) = \frac{1}{12}
\]
Therefore, the probability of rolling a 3 on the red cube and an even number on the blue cube is \(\frac{1}{12}\).