Find the probability of winning the indicated lottery by buying one ticket. In each case, numbers selected are different and order does not matter.

Wisconsin Badger Five Select the five winning numbers from 1 through 31.

To find the probability of winning the Wisconsin Badger Five lottery, where you need to select 5 winning numbers from 1 through 31, and the order of the numbers does not matter, you can use the concept of combinations.

### Steps:

1. **Determine the total number of possible combinations:**
   \[
   \text{Total combinations} = \binom{n}{k}
   \]
   where \( n \) is the total number of numbers to choose from, and \( k \) is the number of numbers to be chosen.

   For the Wisconsin Badger Five:
   \[
   n = 31, \quad k = 5
   \]
   \[
   \text{Total combinations} = \binom{31}{5}
   \]

2. **Calculate the combination:**
   \[
   \binom{31}{5} = \frac{31!}{5! \times (31-5)!} = \frac{31 \times 30 \times 29 \times 28 \times 27}{5 \times 4 \times 3 \times 2 \times 1}
   \]

3. **Simplify the expression:**
   \[
   \binom{31}{5} = \frac{31 \times 30 \times 29 \times 28 \times 27}{120} = 169,911
   \]

4. **Determine the probability:**
   Since only one combination out of all possible combinations is the winning one:
   \[
   \text{Probability of winning} = \frac{1}{169,911}
   \]

### Final Answer:
The probability of winning the Wisconsin Badger Five lottery by buying one ticket is \( \frac{1}{169,911} \) or approximately 0.00000589 (about 0.000589%).

Would you like a detailed explanation or have any questions about this?

### Related Questions:
1. What is the probability of winning a lottery where you choose 6 numbers from 1 through 49?
2. How do you calculate the probability if the order of numbers matters?
3. What is the difference between combinations and permutations in probability?
4. If you bought 10 tickets, what would be your probability of winning?
5. How does increasing the range of numbers affect the probability of winning?

### Tip:
Remember that in combination problems, order does not matter, while in permutation problems, the order does matter.

Learn how to calculate the probability of winning the Wisconsin Badger Five lottery by selecting 5 numbers from 1 to 31. Detailed explanation using combinations and the binomial coefficient formula.

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