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.