Find the probability of winning second prize-that is, picking five of the six winning numbers-with a 6/51 lottery. Write your answer as a fraction.

To find the probability of winning the second prize in a 6/51 lottery, where you need to pick 5 out of the 6 winning numbers, we can use combinatorial probability.

The probability \( P \) can be calculated using the formula:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

### Total Number of Possible Outcomes
The total number of ways to choose 6 numbers from 51 is given by the combination formula:

\[ \binom{51}{6} = \frac{51!}{6!(51-6)!} = \frac{51!}{6! \cdot 45!} \]

### Number of Favorable Outcomes
To win the second prize, we need to match exactly 5 out of the 6 winning numbers. This means we select 5 correct numbers out of the 6 winning numbers, and 1 incorrect number from the remaining 45 numbers.

The number of ways to choose 5 correct numbers out of 6 is:

\[ \binom{6}{5} = 6 \]

The number of ways to choose 1 incorrect number out of the remaining 45 is:

\[ \binom{45}{1} = 45 \]

Therefore, the number of favorable outcomes is:

\[ 6 \times 45 = 270 \]

### Probability Calculation
Now we can calculate the probability:

\[ P = \frac{270}{\binom{51}{6}} = \frac{270}{\frac{51!}{6! \cdot 45!}} \]

We can simplify this by calculating \( \binom{51}{6} \):

\[ \binom{51}{6} = \frac{51 \times 50 \times 49 \times 48 \times 47 \times 46}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

\[ \binom{51}{6} = 18,009,460 \]

So the probability is:

\[ P = \frac{270}{18,009,460} \]

We can simplify this fraction:

\[ P = \frac{27}{1,800,946} \]

Thus, the probability of winning the second prize in a 6/51 lottery is:

\[ \frac{27}{1,800,946} \]

Do you want details or have any questions?

Here are 8 related questions that expand on this information:
1. What is the probability of winning the first prize in a 6/51 lottery?
2. How do you calculate the total number of possible outcomes in a different lottery, such as a 5/60 lottery?
3. What is the probability of picking 4 out of 6 winning numbers in a 6/51 lottery?
4. How does the probability change if the number pool increases from 51 to 60?
5. What are the odds of winning any prize in a typical lottery?
6. How can the combination formula be applied to other real-world problems?
7. How does the factorial function relate to combinations and permutations?
8. Can you explain the difference between permutations and combinations?

**Tip:** Remember, the key to solving probability problems is to carefully determine the total number of possible outcomes and the number of favorable outcomes for the event of interest.

Learn how to calculate the probability of winning the second prize in a 6/51 lottery by picking five of the six winning numbers. This problem involves combinatorial probability and the binomial coefficient formula. Discover the detailed step-by-step solution and related mathematical concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation