Thuy Rolls A Number Cube 7 Times. Which Expression Represents The Probability Of Rolling 5 Successes?${ \begin{array}{c} P(k \text{ Successes }) = {}_n C_k P^k (1-p)^{n-k} \ {}_n C_k = \frac{n!}{(n-k)! \cdot K!} \end{array} } A . \[ A. \[ A . \[ {}_7

Introduction

In probability theory, 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 concept of binomial distribution and how it can be used to calculate the probability of rolling a certain number of successes in a series of independent trials.

The Binomial Distribution Formula

The binomial distribution formula is given by:

$$P(k \text{ successes }) = {}_n C_k p^k (1-p)^{n-k}$$

where:

• $P(k \text{ successes })$ is the probability of getting exactly $k$ successes in $n$ trials
• ${}_n C_k$ is the number of combinations of $n$ items taken $k$ at a time, also known as the binomial coefficient
• $p$ is the probability of success in a single trial
• $n$ is the number of trials
• $k$ is the number of successes

Calculating the Binomial Coefficient

The binomial coefficient ${}_n C_k$ can be calculated using the formula:

$${}_n C_k = \frac{n!}{(n-k)! \cdot k!}$$

where:

• $n!$ is the factorial of $n$, which is the product of all positive integers less than or equal to $n$
• $(n-k)!$ is the factorial of $n-k$
• $k!$ is the factorial of $k$

Thuy's Number Cube Experiment

Thuy rolls a number cube 7 times. We want to find the probability of rolling 5 successes. In this case, the number of trials $n$ is 7, the number of successes $k$ is 5, and the probability of success $p$ is $\frac{1}{6}$, since there are 6 possible outcomes when rolling a fair number cube.

Calculating the Probability

Using the binomial distribution formula, we can calculate the probability of rolling 5 successes as follows:

$$P(5 \text{ successes }) = {}_7 C_5 \left(\frac{1}{6}\right)^5 \left(1-\frac{1}{6}\right)^{7-5}$$

First, we need to calculate the binomial coefficient ${}_7 C_5$:

$${}_7 C_5 = \frac{7!}{(7-5)! \cdot 5!} = \frac{7!}{2! \cdot 5!} = \frac{7 \cdot 6 \cdot 5!}{2 \cdot 5!} = \frac{7 \cdot 6}{2} = 21$$

Now, we can substitute the value of ${}_7 C_5$ into the binomial distribution formula:

$$P(5 \text{ successes }) = 21 \left(\frac{1}{6}\right)^5 \left(1-\frac{1}{6}\right)^{7-5}$$

Simplifying the expression, we get:

$$P(5 \text{ successes }) = 21 \left(\frac{1}{6}\right)^5 \left(\frac{5}{6}\right)^2$$

$$P(5 \text{ successes }) = 21 \left(\frac{1}{6}\right)^5 \left(\frac{25}{36}\right)$$

$$P(5 \text{ successes }) = 21 \left(\frac{1}{7776}\right) \left(\frac{25}{36}\right)$$

$$P(5 \text{ successes }) = 21 \left(\frac{25}{279936}\right)$$

$$P(5 \text{ successes }) = \frac{525}{279936}$$

Conclusion

In this article, we have explored the concept of binomial distribution and how it can be used to calculate the probability of rolling a certain number of successes in a series of independent trials. We have used the binomial distribution formula to calculate the probability of rolling 5 successes in 7 trials, where the probability of success is $\frac{1}{6}$. The result is $\frac{525}{279936}$, which is approximately 0.0187.

Discussion

The binomial distribution is a powerful tool for modeling the number of successes in a fixed number of independent trials. It has many applications in statistics, probability theory, and engineering. In this article, we have used the binomial distribution to calculate the probability of rolling 5 successes in 7 trials. However, the binomial distribution can be used to model many other types of problems, such as the number of defective products in a batch, the number of patients who respond to a treatment, and the number of errors in a computer program.

References

• [1] "Probability and Statistics" by James E. Gentle
• [2] "Binomial Distribution" by Wolfram MathWorld
• [3] "Probability Theory" by E.T. Jaynes

Glossary

• Binomial distribution: a discrete probability distribution that models the number of successes in a fixed number of independent trials
• Binomial coefficient: the number of combinations of $n$ items taken $k$ at a time
• Probability of success: the probability of success in a single trial
• Number of trials: the number of independent trials
• Number of successes: the number of successes in the number of trials
  Thuy Rolls a Number Cube 7 Times: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the concept of binomial distribution and how it can be used to calculate the probability of rolling a certain number of successes in a series of independent trials. We used the binomial distribution formula to calculate the probability of rolling 5 successes in 7 trials, where the probability of success is $\frac{1}{6}$. In this article, we will answer some frequently asked questions about the binomial distribution and its applications.

Q&A

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 is the binomial coefficient?

A: The binomial coefficient is the number of combinations of $n$ items taken $k$ at a time, also known as ${}_n C_k$. It is used in the binomial distribution formula to calculate the probability of getting exactly $k$ successes in $n$ trials.

Q: How do I calculate the binomial coefficient?

A: The binomial coefficient can be calculated using the formula:

$${}_n C_k = \frac{n!}{(n-k)! \cdot k!}$$

where $n!$ is the factorial of $n$, $(n-k)!$ is the factorial of $n-k$, and $k!$ is the factorial of $k$.

Q: What is the probability of success in a single trial?

A: The probability of success in a single trial is denoted by $p$. It is the probability of success in a single trial, and it is used in the binomial distribution formula to calculate the probability of getting exactly $k$ successes in $n$ trials.

Q: How do I calculate the probability of getting exactly $k$ successes in $n$ trials?

A: The probability of getting exactly $k$ successes in $n$ trials can be calculated using the binomial distribution formula:

$$P(k \text{ successes }) = {}_n C_k p^k (1-p)^{n-k}$$

where ${}_n C_k$ is the binomial coefficient, $p$ is the probability of success in a single trial, and $(1-p)$ is the probability of failure in a single trial.

Q: What is the difference between the binomial distribution and the normal distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, while the normal distribution is a continuous probability distribution that models the distribution of a continuous random variable. The binomial distribution is used to model the number of successes in a fixed number of independent trials, while the normal distribution is used to model the distribution of a continuous random variable.

Q: When should I use the binomial distribution?

A: You should use the binomial distribution when you are modeling the number of successes in a fixed number of independent trials, and you know the probability of success in a single trial.

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

A: Some common applications of the binomial distribution include:

• Modeling the number of defective products in a batch
• Modeling the number of patients who respond to a treatment
• Modeling the number of errors in a computer program
• Modeling the number of successes in a series of independent trials

Conclusion

In this article, we have answered some frequently asked questions about the binomial distribution and its applications. We have discussed the binomial distribution formula, the binomial coefficient, and the probability of success in a single trial. We have also discussed some common applications of the binomial distribution and when to use it. We hope that this article has been helpful in understanding the binomial distribution and its applications.

Glossary

• Binomial distribution: a discrete probability distribution that models the number of successes in a fixed number of independent trials
• Binomial coefficient: the number of combinations of $n$ items taken $k$ at a time
• Probability of success: the probability of success in a single trial
• Number of trials: the number of independent trials
• Number of successes: the number of successes in the number of trials

References

• [1] "Probability and Statistics" by James E. Gentle
• [2] "Binomial Distribution" by Wolfram MathWorld
• [3] "Probability Theory" by E.T. Jaynes