Question

Joe has three six‐sided dice that he rolls simultaneously. The faces of the dice contain these letters:

Die #1: A B C D E F

Die #2: G H I J K L

Die #3: M N O P Q R


What is the probability that he will roll three vowels?

Die #1 probability: ?? Die #2 probability: ?? Die #3 probability: ??

Multiply the probabilities to get an answer of: ??

Answer 1

To determine the probability that Joe will roll three vowels, we first need to identify the vowels present on each die.

**Die #1:** A B C D E F
- Vowels: A, E
- Total letters: 6
- Probability of rolling a vowel = Number of vowels / Total letters = 2/6 = 1/3

**Die #2:** G H I J K L
- Vowels: I
- Total letters: 6
- Probability of rolling a vowel = Number of vowels / Total letters = 1/6

**Die #3:** M N O P Q R
- Vowels: O
- Total letters: 6
- Probability of rolling a vowel = Number of vowels / Total letters = 1/6

Now, we multiply the probabilities of each die to find the overall probability of rolling three vowels:

\[
P(\text{three vowels}) = P(\text{vowel on Die #1}) \times P(\text{vowel on Die #2}) \times P(\text{vowel on Die #3})
\]

\[
P(\text{three vowels}) = \frac{1}{3} \times \frac{1}{6} \times \frac{1}{6}
\]

\[
P(\text{three vowels}) = \frac{1}{3} \times \frac{1}{36} = \frac{1}{108}
\]

Thus, the probability that he will roll three vowels is \(\frac{1}{108}\).

To summarize:

- Die #1 probability: \( \frac{1}{3} \)
- Die #2 probability: \( \frac{1}{6} \)
- Die #3 probability: \( \frac{1}{6} \)
- Overall probability for rolling three vowels: \( \frac{1}{108} \)