To solve the problem, we need to find the probability that 6 out of the 7 chosen numbers match the 7 drawn numbers in a lottery where numbers are selected from a pool of 47.

Step-by-Step Solution:

1. Determine the total number of ways to choose 7 numbers from 47.
   This is given by the combination formula: $\binom{47}{7} = \frac{47!}{7!(47-7)!}$

2. Determine the number of ways to match exactly 6 out of the 7 numbers.
   We need to:

   • Choose 6 out of the 7 drawn numbers (which can be done in $\binom{7}{6}$ ways).
   • Choose 1 number from the remaining 40 numbers (which can be done in $\binom{40}{1}$ ways).

   So, the number of ways to have exactly 6 matching numbers is: $\binom{7}{6} \times \binom{40}{1} = 7 \times 40 = 280$

3. Calculate the probability of this event.
   The probability of winning the second prize is given by: $\text{Probability} = \frac{280}{\binom{47}{7}}$

4. Calculate $\binom{47}{7}$. $\binom{47}{7} = \frac{47 \times 46 \times 45 \times 44 \times 43 \times 42 \times 41}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 62,891,499$

5. Simplify the probability. $\frac{280}{62,891,499} = \frac{280}{62,891,499}$

Thus, the probability of winning the second prize is $\frac{280}{62,891,499}$.

Would you like more details on any step, or do you have any questions?

Related Questions:

1. What is the probability of matching all 7 numbers in this lottery game?
2. How would the probability change if the pool was 50 numbers instead of 47?
3. Can you explain why we use combinations rather than permutations in this problem?
4. How do probability and odds differ in lottery games?
5. What strategies do people use to pick numbers in lotteries?

Tip:

When calculating probabilities for lotteries, remember to break the problem into smaller parts: find the total possible outcomes and then count the favorable outcomes.