To solve the lottery probability questions, we can apply combinatorial mathematics.

Part (a): Probability of Picking All 6 Numbers Correctly

1. Total Possible Combinations: The total number of ways to choose 6 different numbers from 45 is given by the combination formula:

   $\text{C}(n, k) = \frac{n!}{k!(n-k)!}$

   Where $n$ is the total numbers available (45) and $k$ is the number of selections (6):

   $\text{C}(45, 6) = \frac{45!}{6!(45-6)!} = \frac{45!}{6! \cdot 39!}$

   This can be computed as:

   $\text{C}(45, 6) = \frac{45 \times 44 \times 43 \times 42 \times 41 \times 40}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 8145060$

2. Probability of Winning: Since there's only one winning combination, the probability $P$ of picking all 6 numbers correctly is:

   $P = \frac{1}{\text{C}(45, 6)} = \frac{1}{8145060} \approx 1.227 \times 10^{-7}$

Part (b): Probability of Picking Exactly 5 of the 6 Numbers Correctly

1. Choosing 5 Correct Numbers: The number of ways to choose 5 correct numbers from the 6 winning numbers:

   $\text{C}(6, 5) = 6$

2. Choosing 1 Incorrect Number: The number of ways to choose 1 incorrect number from the remaining 39 numbers:

   $\text{C}(39, 1) = 39$

3. Total Combinations for Exactly 5 Correct: The total number of ways to choose exactly 5 correct and 1 incorrect number is:

   $\text{C}(6, 5) \times \text{C}(39, 1) = 6 \times 39 = 234$

4. Probability of Picking Exactly 5 Correct Numbers: Thus, the probability $P$ of picking exactly 5 correct numbers is:

   $P = \frac{\text{Number of successful outcomes}}{\text{Total combinations}} = \frac{234}{8145060} \approx 2.872 \times 10^{-5}$

Summary of Results

• Part (a): The probability of picking all 6 numbers correctly is approximately $1.227 \times 10^{-7}$.
• Part (b): The probability of picking exactly 5 of the 6 numbers correctly is approximately $2.872 \times 10^{-5}$.

If you have any questions or need further details, feel free to ask! Here are some related questions for further exploration:

1. How would the probability change if the total numbers were increased to 50?
2. What is the impact of choosing more numbers (like 7 or 8) on the probabilities?
3. How can these calculations help in understanding other lottery formats?
4. What strategies could increase your chances in a lottery?
5. Can you explain the difference between combinations and permutations in this context?

Tip

Always remember to use the factorial function carefully and ensure you understand how combinations work to solve similar probability problems effectively!