) Construct a binomial distribution using n equals 8 and p equals 0.40.

x

​P(x)

0

enter your response here

1

enter your response here

2

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3

enter your response here

4

enter your response here

5

enter your response here

6

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7

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8

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To construct a binomial distribution with \( n = 8 \) and \( p = 0.40 \), we will calculate the probability for each possible outcome \( x \) (where \( x \) ranges from 0 to 8) using the binomial probability formula:

\[
P(x) = \binom{n}{x} p^x (1-p)^{n-x}
\]

Where:
- \( \binom{n}{x} \) is the binomial coefficient, calculated as \( \frac{n!}{x!(n-x)!} \).
- \( p = 0.40 \) is the probability of success on a single trial.
- \( n = 8 \) is the number of trials.

Let's calculate \( P(x) \) for each \( x \) from 0 to 8.

### Calculations:

1. **For \( x = 0 \):**
   \[
   P(0) = \binom{8}{0} (0.40)^0 (0.60)^8 = 1 \times 1 \times 0.60^8 = 0.0168
   \]

2. **For \( x = 1 \):**
   \[
   P(1) = \binom{8}{1} (0.40)^1 (0.60)^7 = 8 \times 0.40 \times 0.60^7 = 0.0896
   \]

3. **For \( x = 2 \):**
   \[
   P(2) = \binom{8}{2} (0.40)^2 (0.60)^6 = 28 \times 0.16 \times 0.60^6 = 0.2090
   \]

4. **For \( x = 3 \):**
   \[
   P(3) = \binom{8}{3} (0.40)^3 (0.60)^5 = 56 \times 0.064 \times 0.60^5 = 0.2787
   \]

5. **For \( x = 4 \):**
   \[
   P(4) = \binom{8}{4} (0.40)^4 (0.60)^4 = 70 \times 0.0256 \times 0.60^4 = 0.2323
   \]

6. **For \( x = 5 \):**
   \[
   P(5) = \binom{8}{5} (0.40)^5 (0.60)^3 = 56 \times 0.01024 \times 0.60^3 = 0.1181
   \]

7. **For \( x = 6 \):**
   \[
   P(6) = \binom{8}{6} (0.40)^6 (0.60)^2 = 28 \times 0.004096 \times 0.60^2 = 0.0368
   \]

8. **For \( x = 7 \):**
   \[
   P(7) = \binom{8}{7} (0.40)^7 (0.60)^1 = 8 \times 0.0016384 \times 0.60 = 0.0079
   \]

9. **For \( x = 8 \):**
   \[
   P(8) = \binom{8}{8} (0.40)^8 (0.60)^0 = 1 \times 0.00065536 \times 1 = 0.0007
   \]

### Binomial Distribution Table

| \( x \) | \( P(x) \) |
|:---:|:-------:|
|  0  |  0.0168 |
|  1  |  0.0896 |
|  2  |  0.2090 |
|  3  |  0.2787 |
|  4  |  0.2323 |
|  5  |  0.1181 |
|  6  |  0.0368 |
|  7  |  0.0079 |
|  8  |  0.0007 |

Would you like more details on how these probabilities are calculated, or do you have any questions?

Here are 5 related questions:
1. How is the binomial coefficient calculated for different values of \( x \)?
2. What is the meaning of the parameters \( n \) and \( p \) in a binomial distribution?
3. How does changing the probability \( p \) affect the binomial distribution?
4. What is the expected value (mean) of this binomial distribution?
5. How can the cumulative distribution function (CDF) be used with a binomial distribution?

**Tip:** The sum of all probabilities in a binomial distribution should always equal 1, which can be a useful check to ensure calculations are correct.

Learn how to construct a binomial distribution with n = 8 and p = 0.40 using the binomial probability formula. Explore the probabilities for each possible outcome from 0 to 8 trials. Suitable for advanced high school students.

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