To solve the problem, we need to find the probability that the number shown on the fourth die (D4) matches at least one of the numbers shown on the first three dice (D1, D2, and D3).
### Step-by-Step Solution:
1. **Identify Total Outcomes**:
Each die has 6 faces, and since there are 4 dice, the total number of outcomes when rolling all four dice is:
\[
\text{Total Outcomes} = 6^4 = 1296
\]
**Hint**: Remember that the total outcomes for multiple dice is the product of the outcomes for each die.
2. **Calculate the Probability of the Complement Event**:
We will first calculate the probability that the number on D4 does not match any of the numbers on D1, D2, or D3.
- If D4 shows a specific number (let's say it shows 'x'), then D1, D2, and D3 must show numbers that are not 'x'.
- Since there are 5 other numbers available (1, 2, 3, 4, 5, or 6 excluding 'x'), each of D1, D2, and D3 can show any of these 5 numbers.
Therefore, the number of favorable outcomes where D4 does not match D1, D2, or D3 is:
\[
\text{Favorable Outcomes} = 5 \times 5 \times 5 \times 6 = 5^3 \times 6 = 125 \times 6 = 750
\]
**Hint**: Consider how many choices are left for D1, D2, and D3 when D4 shows a specific number.
3. **Calculate the Probability of the Complement Event**:
The probability that D4 does not match any of D1, D2, or D3 is given by:
\[
P(\text{D4 does not match D1, D2, D3}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{750}{1296}
\]
**Hint**: The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
4. **Calculate the Required Probability**:
The probability that D4 matches at least one of D1, D2, or D3 is the complement of the above probability:
\[
P(\text{D4 matches D1, D2, or D3}) = 1 - P(\text{D4 does not match D1, D2, D3}) = 1 - \frac{750}{1296}
\]
\[
= \frac{1296 - 750}{1296} = \frac{546}{1296}
\]
5. **Simplify the Probability**:
To simplify \(\frac{546}{1296}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 18:
\[
\frac{546 \div 18}{1296 \div 18} = \frac{91}{216}
\]
**Hint**: Always check if the fraction can be simplified for a cleaner answer.
### Final Answer:
The probability that the number shown on D4 matches at least one of the numbers shown on D1, D2, or D3 is:
\[
\frac{91}{216}
\]
