What Is The Probability Distribution If X ∼ B ( 3 , 0.82 X \sim B(3, 0.82 X ∼ B ( 3 , 0.82 ]?A. ${ \begin{array}{l} P(X=0)=0.03 \ P(X=1)=0.21 \ P(X=2)=0.37 \ P(X=3)=0.61 \end{array} } B . B. B . [ \begin{array}{l} P(X=0)=0.55 \ P(X=1)=0.36 \ P(X=2)=0.08

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Introduction

The 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 explore the binomial distribution with $X \sim B(3, 0.82)$, where $X$ represents the number of successes in 3 independent trials, and the probability of success in each trial is 0.82.

The Binomial Distribution Formula

The binomial distribution is given by the formula:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $n$ is the number of trials
• $k$ is the number of successes
• $p$ is the probability of success in each trial
• $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ successes from $n$ trials

Calculating the Probability Distribution

To calculate the probability distribution of $X \sim B(3, 0.82)$, we need to plug in the values of $n$, $k$, and $p$ into the binomial distribution formula.

For $n=3$, $k=0$, $p=0.82$, we get:

$$P(X=0) = \binom{3}{0} (0.82)^0 (1-0.82)^{3-0} = 0.18^3 = 0.005832$$

However, this is not the correct answer. We need to calculate the probability distribution for each possible value of $X$.

Calculating the Probability Distribution for Each Value of $X$

We need to calculate the probability distribution for each possible value of $X$, which is $X=0$, $X=1$, $X=2$, and $X=3$.

For $X=0$, we have:

$$P(X=0) = \binom{3}{0} (0.82)^0 (1-0.82)^{3-0} = 0.18^3 = 0.005832$$

However, this is not the correct answer. We need to calculate the probability distribution for each possible value of $X$.

For $X=1$, we have:

$$P(X=1) = \binom{3}{1} (0.82)^1 (1-0.82)^{3-1} = 3 \times 0.82 \times 0.18^2 = 0.187056$$

For $X=2$, we have:

$$P(X=2) = \binom{3}{2} (0.82)^2 (1-0.82)^{3-2} = 3 \times 0.82^2 \times 0.18 = 0.365472$$

For $X=3$, we have:

$$P(X=3) = \binom{3}{3} (0.82)^3 (1-0.82)^{3-3} = 0.82^3 = 0.571088$$

Conclusion

In conclusion, the probability distribution of $X \sim B(3, 0.82)$ is given by:

$$P(X=0)=0.005832$$

$$P(X=1)=0.187056$$

$$P(X=2)=0.365472$$

$$P(X=3)=0.571088$$

This is the correct answer.

Discussion

The binomial distribution is a widely used probability distribution in statistics and engineering. It is used to model the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

In this article, we have explored the binomial distribution with $X \sim B(3, 0.82)$, where $X$ represents the number of successes in 3 independent trials, and the probability of success in each trial is 0.82.

We have calculated the probability distribution for each possible value of $X$, which is $X=0$, $X=1$, $X=2$, and $X=3$.

The probability distribution of $X \sim B(3, 0.82)$ is given by:

$$P(X=0)=0.005832$$

$$P(X=1)=0.187056$$

$$P(X=2)=0.365472$$

$$P(X=3)=0.571088$$

This is the correct answer.

References

• [1] Johnson, N. L., Kotz, S., & Kemp, A. W. (1992). Univariate discrete distributions. John Wiley & Sons.
• [2] Feller, W. (1968). An introduction to probability theory and its applications. John Wiley & Sons.

Appendix

The 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.

The binomial distribution is given by the formula:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $n$ is the number of trials
• $k$ is the number of successes
• $p$ is the probability of success in each trial
• $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ successes from $n$ trials

The binomial distribution is widely used in statistics and engineering to model the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

In this article, we have explored the binomial distribution with $X \sim B(3, 0.82)$, where $X$ represents the number of successes in 3 independent trials, and the probability of success in each trial is 0.82.

We have calculated the probability distribution for each possible value of $X$, which is $X=0$, $X=1$, $X=2$, and $X=3$.

The probability distribution of $X \sim B(3, 0.82)$ is given by:

$$P(X=0)=0.005832$$

$$P(X=1)=0.187056$$

$$P(X=2)=0.365472$$

$$P(X=3)=0.571088$$

This is the correct answer.

Glossary

• Binomial distribution: 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.
• Probability of success: The probability of success in each trial.
• Number of trials: The number of independent trials.
• Number of successes: The number of successes in the fixed number of independent trials.
• Binomial coefficient: The number of ways to choose $k$ successes from $n$ trials.

Index

• Binomial distribution: 1-5
• Probability of success: 1-5
• Number of trials: 1-5
• Number of successes: 1-5
• Binomial coefficient: 1-5
  

Q: What is the binomial distribution?

A: The 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.

Q: What are the parameters of the binomial distribution?

A: The parameters of the binomial distribution are:

• $n$: the number of trials
• $p$: the probability of success in each trial
• $k$: the number of successes

Q: How is the binomial distribution calculated?

A: The binomial distribution is calculated using the formula:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $n$ is the number of trials
• $k$ is the number of successes
• $p$ is the probability of success in each trial
• $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ successes from $n$ trials

Q: What is the binomial coefficient?

A: The binomial coefficient is the number of ways to choose $k$ successes from $n$ trials. It is calculated using the formula:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Q: What is the probability of success in each trial?

A: The probability of success in each trial is denoted by $p$. It is a value between 0 and 1 that represents the probability of success in each trial.

Q: What is the number of trials?

A: The number of trials is denoted by $n$. It is the number of independent trials that are conducted.

Q: What is the number of successes?

A: The number of successes is denoted by $k$. It is the number of successes in the fixed number of independent trials.

Q: How is the binomial distribution used in real-life applications?

A: The binomial distribution is widely used in real-life applications, such as:

• Quality control: to model the number of defective products in a batch
• Medical research: to model the number of patients who respond to a treatment
• Finance: to model the number of successful investments in a portfolio
• Engineering: to model the number of failures in a system

Q: What are the assumptions of the binomial distribution?

A: The assumptions of the binomial distribution are:

• Independence: each trial is independent of the others
• Constant probability of success: the probability of success in each trial is constant
• Fixed number of trials: the number of trials is fixed

Q: What are the limitations of the binomial distribution?

A: The limitations of the binomial distribution are:

• Discrete distribution: the binomial distribution is a discrete distribution, which means it can only take on specific values
• Assumes independence: the binomial distribution assumes that each trial is independent of the others, which may not always be the case
• Assumes constant probability of success: the binomial distribution assumes that the probability of success in each trial is constant, which may not always be the case

Q: How is the binomial distribution related to other distributions?

A: The binomial distribution is related to other distributions, such as:

• Normal distribution: the binomial distribution can be approximated by the normal distribution for large values of $n$
• Poisson distribution: the binomial distribution can be approximated by the Poisson distribution for large values of $n$ and small values of $p$

Q: What are some common applications of the binomial distribution?

A: Some common applications of the binomial distribution include:

• Quality control: to model the number of defective products in a batch
• Medical research: to model the number of patients who respond to a treatment
• Finance: to model the number of successful investments in a portfolio
• Engineering: to model the number of failures in a system

Q: How can I calculate the binomial distribution using a calculator or computer software?

A: You can calculate the binomial distribution using a calculator or computer software, such as:

• Microsoft Excel: you can use the BINOM.DIST function to calculate the binomial distribution
• R: you can use the dbinom function to calculate the binomial distribution
• Python: you can use the scipy.stats.binom function to calculate the binomial distribution

Q: What are some common mistakes to avoid when using the binomial distribution?

A: Some common mistakes to avoid when using the binomial distribution include:

• Assuming independence: assuming that each trial is independent of the others, when in fact they may not be
• Assuming constant probability of success: assuming that the probability of success in each trial is constant, when in fact it may not be
• Using the binomial distribution for non-discrete data: using the binomial distribution for non-discrete data, when in fact it is a discrete distribution.