Zaheer Khan has taken 5 five-wicket hauls in his last 17 matches. His match records are selected at random, one by one, and analyzed. If none of the match records is analyzed more than once, then what is the probability that the 9th one analyzed is his last five-wicket haul match?

To determine the probability that the 9th match analyzed is Zaheer Khan's last five-wicket haul match, we can approach the problem step-by-step.

First, we recognize the following:
- Zaheer Khan has taken 5 five-wicket hauls in his last 17 matches.
- We are selecting these 17 matches one by one at random.

We want to calculate the probability that the 9th match analyzed is the 5th (and last) five-wicket haul. This means, in the first 8 matches analyzed, there must be exactly 4 five-wicket hauls.

Here’s the structured solution:

1. **Total ways to choose the 8 matches from 17:**
   \[
   \binom{17}{8}
   \]

2. **Ways to choose 4 five-wicket hauls out of the first 8 matches:**
   \[
   \binom{5}{4}
   \]

3. **Ways to choose the remaining 4 matches (non-five-wicket hauls) out of the remaining 12 matches:**
   \[
   \binom{12}{4}
   \]

Combining the two, the number of favorable ways:
\[
\binom{5}{4} \times \binom{12}{4}
\]

4. **Probability:**
   The total number of ways to choose 8 matches out of 17 is:
   \[
   \binom{17}{8}
   \]

   So, the probability \( P \) is:
   \[
   P = \frac{\binom{5}{4} \times \binom{12}{4}}{\binom{17}{8}}
   \]

5. **Calculations:**
   \[
   \binom{5}{4} = 5
   \]
   \[
   \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \times 8!} = 495
   \]
   \[
   \binom{17}{8} = \frac{17!}{8! \times 9!} = 24310
   \]

   Therefore, the probability is:
   \[
   P = \frac{5 \times 495}{24310} = \frac{2475}{24310}
   \]

6. **Simplify the fraction:**
   \[
   \frac{2475}{24310} = \frac{2475 \div 5}{24310 \div 5} = \frac{495}{4862}
   \]
   Simplifying further:
   \[
   \frac{495}{4862} = \frac{99}{972.4} \approx 0.1018
   \]

Thus, the probability that the 9th match analyzed is his last five-wicket haul match is approximately 0.1018, or about 10.18%.

Would you like more details or have any questions? 

Here are 8 related questions to expand your understanding:
1. What is a binomial coefficient and how is it calculated?
2. How can the binomial theorem be applied in probability problems?
3. What are some other examples of combinatorial probability problems?
4. How do probability and combinatorics intersect in real-world scenarios?
5. What is the difference between permutations and combinations?
6. How does the inclusion-exclusion principle apply in probability?
7. What is the importance of sample space in probability theory?
8. How can we use probability to make predictions in sports statistics?

**Tip:** When solving probability problems involving combinations, always ensure to carefully count both favorable and total possible outcomes to get accurate results.

Calculate the probability that the 9th match analyzed is Zaheer Khan's last five-wicket haul match using combinatorics and probability. Understand how binomial coefficients are applied in this problem. Suitable for Grades 10-12.

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