To calculate the probability of winning the second prize in this lottery game, where the player picks 9 numbers and must match 8 out of 9, follow these steps:

1. Total number of possible outcomes: The player picks 9 numbers from a set of 51. The number of possible ways to choose 9 numbers from 51 is given by the combination:

   $\binom{51}{9} = \frac{51!}{9!(51-9)!} = \binom{51}{9}$

2. Favorable outcomes: To win second prize, the player needs 8 of their 9 selected numbers to match the drawn numbers. This means 8 of the 9 chosen numbers must be correct, and 1 must be incorrect. The number of favorable outcomes involves:

   • Choosing 8 correct numbers out of the 9 correct ones from the drawn numbers (which is $\binom{9}{8}$).
   • Choosing 1 incorrect number from the 42 remaining numbers (since there are 51 numbers in total, and 9 have been drawn). This is given by $\binom{42}{1}$.

   Thus, the number of favorable outcomes is:

   $\binom{9}{8} \times \binom{42}{1} = 9 \times 42$

3. Probability: The probability of winning the second prize is the ratio of favorable outcomes to total outcomes:

   $\text{Probability} = \frac{\binom{9}{8} \times \binom{42}{1}}{\binom{51}{9}} = \frac{9 \times 42}{\binom{51}{9}}$

Now, let’s compute this probability.

I'll calculate it for you.The number of favorable outcomes is 378, while the total number of possible outcomes is 3,042,312,350. Therefore, the probability of winning the second prize is:

$\frac{378}{3,042,312,350}$

This is the exact probability of winning the second prize.

Would you like more details or clarification on any part of the process? Here are five related questions you might find interesting:

1. How is the probability affected if the total numbers increase to 60 instead of 51?
2. What is the probability of matching exactly 7 out of 9 numbers?
3. How can we express this probability in decimal form?
4. What is the expected number of second prize winners in a large number of players?
5. What are the odds of winning the jackpot (all 9 numbers correct)?

Tip: Understanding how combinations work is key to solving lottery probability problems efficiently.