Look at this:


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Part 1

A state lottery game requires that you pick 6 different numbers from 1 to 

4343.

​(a) Assuming that order is​ unimportant, what is the probability of picking all 6 numbers correctly to win the big​ prize?

​(b) What is the probability of picking exactly 

55 

of the 6 numbers​ correctly?

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Part 1

​(a) The probability of picking all 6 numbers drawn is 

1.640 times 10 Superscript negative 71.640×10−7.

​(Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as​ needed.) 

Part 2

​(b) The probability of selecting exactly 

55 

of the 6 numbers drawn is 

3.641 times 10 Superscript negative 53.641×10−5.

​(Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as​ needed.) 

figure out how those were solved to solve this one:


Question content area top

Part 1

A state lottery game requires that you pick 6 different numbers from 1 to 

4545.

​(a) Assuming that order is​ unimportant, what is the probability of picking all 6 numbers correctly to win the big​ prize?

​(b) What is the probability of picking exactly 

55 

of the 6 numbers​ correctly?

Question content area bottom

Part 1

​(a) The probability of picking all 6 numbers drawn is 

enter your response here.

​(Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as​ needed.) 


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!

A state lottery game requires that you pick 6 different numbers from 1 to 45. (a) What is the probability of picking all 6 numbers correctly to win the big prize? (b) What is the probability of picking exactly 5 of the 6 numbers correctly?

Explore the probability of winning a state lottery by correctly picking 6 numbers from 45. This detailed solution covers both the probability of picking all 6 numbers and exactly 5 numbers correctly, using combinatorial mathematics.

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