Use $n=9$ And $p=0.2$ To Complete Parts (a) Through (d) Below.(a) Construct A Binomial Probability Distribution With The Given Parameters.$\[ \begin{array}{cc} x & P(x) \\ \hline 0 & \square \\ 1 & \square \\ 2 & \square \\ 3

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Introduction

In probability theory, a binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this article, we will use the given parameters $n=9$ and $p=0.2$ to construct a binomial probability distribution and complete parts (a) through (d) below.

Part (a): Construct a Binomial Probability Distribution

To construct a binomial probability distribution, we need to calculate the probability of each possible outcome, which is given by the formula:

P(x)=(nx)px(1βˆ’p)nβˆ’xP(x) = \binom{n}{x} p^x (1-p)^{n-x}

where $n$ is the number of trials, $x$ is the number of successes, $p$ is the probability of success, and $\binom{n}{x}$ is the binomial coefficient.

Using the given parameters $n=9$ and $p=0.2$, we can calculate the probability of each possible outcome as follows:

x P(x)
0
1
2
3

To calculate the probability of each outcome, we need to use the formula above. Let's calculate the probability of each outcome:

  • For $x=0$, we have:

P(0)=(90)(0.2)0(1βˆ’0.2)9βˆ’0P(0) = \binom{9}{0} (0.2)^0 (1-0.2)^{9-0}

P(0)=1Γ—1Γ—(0.8)9P(0) = 1 \times 1 \times (0.8)^9

P(0)=(0.8)9P(0) = (0.8)^9

  • For $x=1$, we have:

P(1)=(91)(0.2)1(1βˆ’0.2)9βˆ’1P(1) = \binom{9}{1} (0.2)^1 (1-0.2)^{9-1}

P(1)=9Γ—0.2Γ—(0.8)8P(1) = 9 \times 0.2 \times (0.8)^8

P(1)=1.8Γ—(0.8)8P(1) = 1.8 \times (0.8)^8

  • For $x=2$, we have:

P(2)=(92)(0.2)2(1βˆ’0.2)9βˆ’2P(2) = \binom{9}{2} (0.2)^2 (1-0.2)^{9-2}

P(2)=36Γ—0.04Γ—(0.8)7P(2) = 36 \times 0.04 \times (0.8)^7

P(2)=1.44Γ—(0.8)7P(2) = 1.44 \times (0.8)^7

  • For $x=3$, we have:

P(3)=(93)(0.2)3(1βˆ’0.2)9βˆ’3P(3) = \binom{9}{3} (0.2)^3 (1-0.2)^{9-3}

P(3)=84Γ—0.008Γ—(0.8)6P(3) = 84 \times 0.008 \times (0.8)^6

P(3)=0.672Γ—(0.8)6P(3) = 0.672 \times (0.8)^6

Now that we have calculated the probability of each outcome, we can construct the binomial probability distribution as follows:

x P(x)
0 (0.8)^9
1 1.8 \times (0.8)^8
2 1.44 \times (0.8)^7
3 0.672 \times (0.8)^6

Part (b): Calculate the Mean and Variance

The mean of a binomial distribution is given by the formula:

ΞΌ=np\mu = np

where $n$ is the number of trials and $p$ is the probability of success.

Using the given parameters $n=9$ and $p=0.2$, we can calculate the mean as follows:

ΞΌ=9Γ—0.2\mu = 9 \times 0.2

ΞΌ=1.8\mu = 1.8

The variance of a binomial distribution is given by the formula:

Οƒ2=np(1βˆ’p)\sigma^2 = np(1-p)

Using the given parameters $n=9$ and $p=0.2$, we can calculate the variance as follows:

Οƒ2=9Γ—0.2Γ—(1βˆ’0.2)\sigma^2 = 9 \times 0.2 \times (1-0.2)

Οƒ2=9Γ—0.2Γ—0.8\sigma^2 = 9 \times 0.2 \times 0.8

Οƒ2=1.44\sigma^2 = 1.44

Part (c): Calculate the Standard Deviation

The standard deviation of a binomial distribution is the square root of the variance. Using the variance calculated in part (b), we can calculate the standard deviation as follows:

Οƒ=1.44\sigma = \sqrt{1.44}

Οƒ=1.2\sigma = 1.2

Part (d): Calculate the Probability of a Specific Outcome

To calculate the probability of a specific outcome, we can use the formula:

P(x)=(nx)px(1βˆ’p)nβˆ’xP(x) = \binom{n}{x} p^x (1-p)^{n-x}

Using the given parameters $n=9$ and $p=0.2$, we can calculate the probability of a specific outcome as follows:

For example, let's calculate the probability of $x=2$:

P(2)=(92)(0.2)2(1βˆ’0.2)9βˆ’2P(2) = \binom{9}{2} (0.2)^2 (1-0.2)^{9-2}

P(2)=36Γ—0.04Γ—(0.8)7P(2) = 36 \times 0.04 \times (0.8)^7

P(2)=1.44Γ—(0.8)7P(2) = 1.44 \times (0.8)^7

P(2)=0.0945P(2) = 0.0945

Therefore, the probability of $x=2$ is 0.0945.

Conclusion

Q&A: Binomial Probability Distribution

Q: What is a binomial probability distribution?

A: A binomial probability distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

Q: What are the parameters of a binomial probability distribution?

A: The parameters of a binomial probability distribution are:

  • n: The number of trials
  • p: The probability of success

Q: How is the probability of each outcome calculated in a binomial probability distribution?

A: The probability of each outcome is calculated using the formula:

P(x)=(nx)px(1βˆ’p)nβˆ’xP(x) = \binom{n}{x} p^x (1-p)^{n-x}

where $n$ is the number of trials, $x$ is the number of successes, $p$ is the probability of success, and $\binom{n}{x}$ is the binomial coefficient.

Q: What is the mean of a binomial probability distribution?

A: The mean of a binomial probability distribution is given by the formula:

ΞΌ=np\mu = np

where $n$ is the number of trials and $p$ is the probability of success.

Q: What is the variance of a binomial probability distribution?

A: The variance of a binomial probability distribution is given by the formula:

Οƒ2=np(1βˆ’p)\sigma^2 = np(1-p)

Q: What is the standard deviation of a binomial probability distribution?

A: The standard deviation of a binomial probability distribution is the square root of the variance.

Q: How is the probability of a specific outcome calculated in a binomial probability distribution?

A: The probability of a specific outcome is calculated using the formula:

P(x)=(nx)px(1βˆ’p)nβˆ’xP(x) = \binom{n}{x} p^x (1-p)^{n-x}

Q: What are some real-world applications of binomial probability distributions?

A: Binomial probability distributions have many real-world applications, including:

  • Quality control: Binomial probability distributions are used to model the number of defective products in a batch.
  • Medical research: Binomial probability distributions are used to model the number of patients who respond to a treatment.
  • Finance: Binomial probability distributions are used to model the number of successful investments in a portfolio.

Q: How can I use binomial probability distributions in my own work or research?

A: To use binomial probability distributions in your own work or research, you can:

  • Identify the parameters: Determine the number of trials and the probability of success.
  • Calculate the probabilities: Use the formula to calculate the probability of each outcome.
  • Analyze the results: Use the probabilities to make informed decisions or predictions.

Conclusion

In this article, we have provided a comprehensive guide to binomial probability distributions, including a Q&A section to help you understand the concepts and applications. We hope this article has been helpful in your understanding of binomial probability distributions.