Question
What is the probability of drawing the correct sequence of 5 numbers in a lottery where 6 different numbers are drawn between 1 and 32 inclusive?
Answers
GPT-4o mini
To find the probability of drawing the correct sequence of 5 numbers in a lottery where 6 different numbers are drawn from a set of numbers ranging from 1 to 32, we need to clarify the conditions of the problem.
Assuming that you need to select 5 specific numbers from a total of 6 numbers drawn (with the remaining number not affecting your selection), let's break down the probability calculation:
1. **Total Numbers**: There are 32 numbers in total (labeled 1 through 32).
2. **Number of Draws**: 6 numbers are drawn from these 32.
3. **Correct Sequence**: You need to choose 5 out of the 6 drawn numbers in the correct order.
### Steps to Calculate the Probability:
1. **Ways to Choose 5 out of 6**: The order matters, so you will calculate the number of ways to choose and arrange 5 specific numbers out of the 6 drawn.
2. **Total Possible Sequences of 5**: The total distinct arrangements of drawing 5 numbers from the 32 possible could be calculated using permutations but since you are focusing on 6 drawn numbers, we consider:
- You have to select 5 numbers from the 6, and they need to be in a specific order.
The number of ways to choose 5 numbers from 6 is calculated as \( P(6, 5) \):
\[
P(6, 5) = 6! / (6-5)! = 6! / 1! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 = 720
\]
3. **Total Possible Outcomes**: The total number of ways to select 6 numbers from 32 is computed through combinations (since order does not matter in the draw):
\[
C(32, 6) = \frac{32!}{6!(32-6)!}
\]
4. **Probability Calculation**: The probability of drawing those correct 5 numbers in the right order from any of the total outcomes is defined as:
\[
P(\text{correct sequence}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}}
\]
Where:
- **Number of Favorable Outcomes** = 720 (the arrangements of 5 out of the 6 drawn)
- **Total Outcomes** = \( C(32, 6) \)
### Calculation of \( C(32, 6) \):
Calculating \( C(32, 6) \):
\[
C(32, 6) = \frac{32 \times 31 \times 30 \times 29 \times 28 \times 27}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{32!}{6! \times 26!} = \frac{32 \times 31 \cdots 27}{720}
\]
Calculating the numerator:
\[
32 \times 31 = 992, \quad 992 \times 30 = 29760, \quad 29760 \times 29 = 863040, \quad 863040 \times 28 = 24192000, \quad 24192000 \times 27 = 653184000
\]
Thus:
\[
C(32, 6) = \frac{653184000}{720} = 90720
\]
5. **Final Calculation of Probability**:
Combining all,
\[
P(\text{correct sequence}) = \frac{720}{90720} = \frac{1}{126}.
\]
Thus, the probability of drawing the correct sequence of 5 numbers out of a total of 6 drawn is \( \frac{1}{126} \).
Assuming that you need to select 5 specific numbers from a total of 6 numbers drawn (with the remaining number not affecting your selection), let's break down the probability calculation:
1. **Total Numbers**: There are 32 numbers in total (labeled 1 through 32).
2. **Number of Draws**: 6 numbers are drawn from these 32.
3. **Correct Sequence**: You need to choose 5 out of the 6 drawn numbers in the correct order.
### Steps to Calculate the Probability:
1. **Ways to Choose 5 out of 6**: The order matters, so you will calculate the number of ways to choose and arrange 5 specific numbers out of the 6 drawn.
2. **Total Possible Sequences of 5**: The total distinct arrangements of drawing 5 numbers from the 32 possible could be calculated using permutations but since you are focusing on 6 drawn numbers, we consider:
- You have to select 5 numbers from the 6, and they need to be in a specific order.
The number of ways to choose 5 numbers from 6 is calculated as \( P(6, 5) \):
\[
P(6, 5) = 6! / (6-5)! = 6! / 1! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 = 720
\]
3. **Total Possible Outcomes**: The total number of ways to select 6 numbers from 32 is computed through combinations (since order does not matter in the draw):
\[
C(32, 6) = \frac{32!}{6!(32-6)!}
\]
4. **Probability Calculation**: The probability of drawing those correct 5 numbers in the right order from any of the total outcomes is defined as:
\[
P(\text{correct sequence}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}}
\]
Where:
- **Number of Favorable Outcomes** = 720 (the arrangements of 5 out of the 6 drawn)
- **Total Outcomes** = \( C(32, 6) \)
### Calculation of \( C(32, 6) \):
Calculating \( C(32, 6) \):
\[
C(32, 6) = \frac{32 \times 31 \times 30 \times 29 \times 28 \times 27}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{32!}{6! \times 26!} = \frac{32 \times 31 \cdots 27}{720}
\]
Calculating the numerator:
\[
32 \times 31 = 992, \quad 992 \times 30 = 29760, \quad 29760 \times 29 = 863040, \quad 863040 \times 28 = 24192000, \quad 24192000 \times 27 = 653184000
\]
Thus:
\[
C(32, 6) = \frac{653184000}{720} = 90720
\]
5. **Final Calculation of Probability**:
Combining all,
\[
P(\text{correct sequence}) = \frac{720}{90720} = \frac{1}{126}.
\]
Thus, the probability of drawing the correct sequence of 5 numbers out of a total of 6 drawn is \( \frac{1}{126} \).