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? (8 points; 2 per answer)

Die #1 probability: 13 Die #2 probability: 16 Die #3 probability: 16

Multiply the probabilities to get an answer of: 1108
What is the probability that he will roll the word DIP? (8 points; 2 per answer)

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 using the three dice with their respective letters, let's first identify the vowels present on each die:

- Die #1: A, B, C, D, E, F (vowels: A, E)
- Die #2: G, H, I, J, K, L (vowels: I)
- Die #3: M, N, O, P, Q, R (vowels: O)

### 1. Probability of rolling three vowels:

- **Die #1:**
- There are 2 vowels (A, E) on Die #1 out of 6 faces.
- Probability of rolling a vowel = \( \frac{2}{6} = \frac{1}{3} \)

- **Die #2:**
- There is 1 vowel (I) on Die #2 out of 6 faces.
- Probability of rolling a vowel = \( \frac{1}{6} \)

- **Die #3:**
- There is 1 vowel (O) on Die #3 out of 6 faces.
- Probability of rolling a vowel = \( \frac{1}{6} \)

### Total Probability of Rolling Three Vowels:
Now, we multiply the probabilities of rolling a vowel on each die:

\[
\text{Probability of three vowels} = \left(\frac{1}{3}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) = \frac{1}{3} \times \frac{1}{36} = \frac{1}{108}
\]

### 2. Probability of rolling the word DIP:
Next, we find the probability of rolling the specific letters D, I, and P from the respective dice.

- **Die #1 (to roll D):**
- There is 1 face (D) out of 6 faces.
- Probability of rolling D = \( \frac{1}{6} \)

- **Die #2 (to roll I):**
- There is 1 face (I) out of 6 faces.
- Probability of rolling I = \( \frac{1}{6} \)

- **Die #3 (to roll P):**
- There is 1 face (P) out of 6 faces.
- Probability of rolling P = \( \frac{1}{6} \)

### Total Probability of Rolling DIP:
Now, we multiply the probabilities of rolling D, I, and P on their respective dice:
\[
\text{Probability of rolling DIP} = \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) = \frac{1}{216}
\]

### Summary of Answers:
1. Probability of rolling three vowels:
- Die #1 probability: \( \frac{1}{3} \)
- Die #2 probability: \( \frac{1}{6} \)
- Die #3 probability: \( \frac{1}{6} \)
- Multiply: \( \frac{1}{108} \)

2. Probability of rolling the word DIP:
- Die #1 probability: \( \frac{1}{6} \)
- Die #2 probability: \( \frac{1}{6} \)
- Die #3 probability: \( \frac{1}{6} \)
- Multiply: \( \frac{1}{216} \)