Roll a fair die 6 times. Obtain the probability of at least one place match.

Steps to Solve

1. Define the complement of the desired outcome

To find the probability of getting at least one matching number when rolling a die six times, it can be easier to first calculate the complement, which is the probability of getting no matching numbers at all.

2. Calculate the total number of outcomes

When rolling a die once, there are 6 possible outcomes. For six rolls, the total number of outcomes is:

$$ 6^6 $$

3. Calculate the number of successful outcomes for no matches

To calculate the number of ways to roll the die six times with no matching numbers, note that the first roll can be any of the 6 numbers, the second roll can be one of the remaining 5 numbers, the third can be one of the remaining 4 numbers, and so on. Thus, the number of successful outcomes is:

$$ 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! $$

4. Calculate the probability of no matches

Now, the probability of getting no matching numbers is the number of successful outcomes divided by the total outcomes:

$$ P(\text{no matches}) = \frac{6!}{6^6} $$

5. Calculate the probability of at least one match

Finally, to find the probability of getting at least one matching number, we subtract the probability of no matches from 1:

$$ P(\text{at least one match}) = 1 - P(\text{no matches}) $$