lily has three fair dice: one 4-side, one 6-side and one 8-side. Lily picks two dice at random and rolls them. Given that both dice roll the same number, what is the probability that she rolls the third die, she obtains that same number again? If the answer can be expressed as m/n, where m and n are relatively prime positive integers. find m+n

Let's break down the problem step by step.

### Step 1: Understanding the Problem
Lily has three dice:
- A 4-sided die (D4)
- A 6-sided die (D6)
- An 8-sided die (D8)

She picks two dice at random and rolls them. Given that both dice roll the same number, we need to find the probability that if she rolls the third die, it shows that same number.

### Step 2: Determine the Possible Pairs
The possible pairs of dice Lily can choose are:
1. D4 and D6
2. D4 and D8
3. D6 and D8

### Step 3: Calculate the Probability of Rolling the Same Number for Each Pair
We need to determine the probability of rolling the same number on both dice for each pair:

1. **D4 and D6**:
   - The possible outcomes are {1, 2, 3, 4} for D4 and {1, 2, 3, 4, 5, 6} for D6.
   - The favorable outcomes are {1, 2, 3, 4}.
   - Probability of rolling the same number: 
     \[
     P(\text{same number for D4 and D6}) = \frac{4}{4 \times 6} = \frac{1}{6}
     \]

2. **D4 and D8**:
   - The possible outcomes are {1, 2, 3, 4} for D4 and {1, 2, 3, 4, 5, 6, 7, 8} for D8.
   - The favorable outcomes are {1, 2, 3, 4}.
   - Probability of rolling the same number: 
     \[
     P(\text{same number for D4 and D8}) = \frac{4}{4 \times 8} = \frac{1}{8}
     \]

3. **D6 and D8**:
   - The possible outcomes are {1, 2, 3, 4, 5, 6} for D6 and {1, 2, 3, 4, 5, 6, 7, 8} for D8.
   - The favorable outcomes are {1, 2, 3, 4, 5, 6}.
   - Probability of rolling the same number: 
     \[
     P(\text{same number for D6 and D8}) = \frac{6}{6 \times 8} = \frac{1}{8}
     \]

### Step 4: Calculate the Conditional Probability for the Third Die
Given that the two dice show the same number, we need to calculate the probability that the third die, when rolled, also shows the same number.

**Case 1: D4 and D6 were rolled**  
If D4 and D6 show the same number, the possible numbers are {1, 2, 3, 4}. If D8 is the third die:
\[
P(\text{D8 matches the number}) = \frac{4}{8} = \frac{1}{2}
\]

**Case 2: D4 and D8 were rolled**  
If D4 and D8 show the same number, the possible numbers are {1, 2, 3, 4}. If D6 is the third die:
\[
P(\text{D6 matches the number}) = \frac{4}{6} = \frac{\frac{2}{3}}
\]

**Case 3: D6 and D8 were rolled**  
If D6 and D8 show the same number, the possible numbers are {1, 2, 3, 4, 5, 6}. If D4 is the third die:
\[
P(\text{D4 matches the number}) = \frac{4}{6} = \frac{2}{3}
\]

### Step 5: Combine All Cases Using Total Probability
We will use the law of total probability to find the final probability. We have:
\[
P(\text{same number on all three}) = P(\text{D4 and D6}) \cdot P(\text{D8 matches}) + P(\text{D4 and D8}) \cdot P(\text{D6 matches}) + P(\text{D6 and D8}) \cdot P(\text{D4 matches})
\]
Substituting the values:
\[
P(\text{same number on all three}) = \left(\frac{1}{3} \cdot \frac{1}{6}\right) \cdot \frac{1}{2} + \left(\frac{1}{3} \cdot \frac{1}{8}\right) \cdot \frac{2}{3} + \left(\frac{1}{3} \cdot \frac{1}{8}\right) \cdot \frac{2}{3}
\]
\[
P(\text{same number on all three}) = \frac{1}{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{2} + \frac{1}{8} \cdot \frac{2}{3} + \frac{1}{8} \cdot \frac{2}{3} \right)
\]
Simplifying further:
\[
P(\text{same number on all three}) = \frac{1}{3} \cdot \left(\frac{1}{12} + \frac{1}{12} + \frac{1}{12}\right) = \frac{1}{3} \cdot \frac{3}{12} = \frac{1}{4}
\]

### Final Step: Answer
The required probability is \( \frac{1}{4} \). The answer can be expressed as \( m/n = 1/4 \) where \( m = 1 \) and \( n = 4 \), so \( m+n = 1+4 = 5 \).

**Final Answer:** \( \boxed{5} \)

Explore a probability problem involving three dice (4-sided, 6-sided, and 8-sided) where Lily rolls two dice and looks for the probability that the third die shows the same number. Understand conditional probability and its application in dice games.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation