Which Value For $x$ Will Result In The Greatest Relative Frequency Given The Discrete Random Variable $X$, When $X \sim B\left(4, \frac{1}{10}\right)$?A. $ X = 1 X=1 X = 1 [/tex] B. $x=0$ C.

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Introduction

In probability theory, a discrete random variable is a variable that can take on a countable number of distinct values. 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 relative frequency and determine which value of $x$ will result in the greatest relative frequency given the discrete random variable $X$, where $X \sim B\left(4, \frac{1}{10}\right)$.

What is Relative Frequency?

Relative frequency is a measure of the proportion of times a particular value occurs in a sample or population. In the context of a discrete random variable, relative frequency is the probability of observing a particular value. For example, if we roll a fair six-sided die, the relative frequency of rolling a 4 is 1/6, since there is one favorable outcome (rolling a 4) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).

The Binomial Distribution

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 probability mass function (PMF) of the binomial distribution is given by:

P(X=k)=(nk)pk(1−p)n−kP(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 on each trial

  • \binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ successes from $n$ trials

The Problem

We are given a discrete random variable $X$, where $X \sim B\left(4, \frac{1}{10}\right)$. This means that $X$ represents the number of successes in 4 independent trials, where each trial has a probability of success of $\frac{1}{10}$. We want to determine which value of $x$ will result in the greatest relative frequency.

Calculating Relative Frequency

To calculate the relative frequency of each value of $x$, we need to calculate the probability of each value using the PMF of the binomial distribution. We will calculate the probability of each value of $x$ from 0 to 4.

Calculating the Probability of $x = 0$

The probability of $x = 0$ is given by:

P(X=0)=(40)(110)0(1−110)4−0P(X = 0) = \binom{4}{0} \left(\frac{1}{10}\right)^0 \left(1-\frac{1}{10}\right)^{4-0}

P(X=0)=(910)4P(X = 0) = \left(\frac{9}{10}\right)^4

Calculating the Probability of $x = 1$

The probability of $x = 1$ is given by:

P(X=1)=(41)(110)1(1−110)4−1P(X = 1) = \binom{4}{1} \left(\frac{1}{10}\right)^1 \left(1-\frac{1}{10}\right)^{4-1}

P(X=1)=4(110)(910)3P(X = 1) = 4 \left(\frac{1}{10}\right) \left(\frac{9}{10}\right)^3

Calculating the Probability of $x = 2$

The probability of $x = 2$ is given by:

P(X=2)=(42)(110)2(1−110)4−2P(X = 2) = \binom{4}{2} \left(\frac{1}{10}\right)^2 \left(1-\frac{1}{10}\right)^{4-2}

P(X=2)=6(1100)(910)2P(X = 2) = 6 \left(\frac{1}{100}\right) \left(\frac{9}{10}\right)^2

Calculating the Probability of $x = 3$

The probability of $x = 3$ is given by:

P(X=3)=(43)(110)3(1−110)4−3P(X = 3) = \binom{4}{3} \left(\frac{1}{10}\right)^3 \left(1-\frac{1}{10}\right)^{4-3}

P(X=3)=4(11000)(910)1P(X = 3) = 4 \left(\frac{1}{1000}\right) \left(\frac{9}{10}\right)^1

Calculating the Probability of $x = 4$

The probability of $x = 4$ is given by:

P(X=4)=(44)(110)4(1−110)4−4P(X = 4) = \binom{4}{4} \left(\frac{1}{10}\right)^4 \left(1-\frac{1}{10}\right)^{4-4}

P(X=4)=(110000)P(X = 4) = \left(\frac{1}{10000}\right)

Comparing Relative Frequencies

Now that we have calculated the probability of each value of $x$, we can compare the relative frequencies to determine which value of $x$ will result in the greatest relative frequency.

Value of $x$ Probability
0 $\left(\frac{9}{10}\right)^4$
1 $4 \left(\frac{1}{10}\right) \left(\frac{9}{10}\right)^3$
2 $6 \left(\frac{1}{100}\right) \left(\frac{9}{10}\right)^2$
3 $4 \left(\frac{1}{1000}\right) \left(\frac{9}{10}\right)^1$
4 $\left(\frac{1}{10000}\right)$

From the table, we can see that the probability of $x = 0$ is the greatest, followed by the probability of $x = 1$. Therefore, the value of $x$ that will result in the greatest relative frequency is $x = 0$.

Conclusion

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 probability mass function (PMF) of the binomial distribution?

A: The PMF of the binomial distribution is given by:

P(X=k)=(nk)pk(1−p)n−kP(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 on each trial

  • \binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ successes from $n$ trials

Q: How do I calculate the probability of each value of $x$ in the binomial distribution?

A: To calculate the probability of each value of $x$, you need to use the PMF of the binomial distribution. You can use the formula:

P(X=k)=(nk)pk(1−p)n−kP(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 on 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 relative frequency?

A: Relative frequency is a measure of the proportion of times a particular value occurs in a sample or population. In the context of a discrete random variable, relative frequency is the probability of observing a particular value.

Q: How do I compare the relative frequencies of different values of $x$?

A: To compare the relative frequencies of different values of $x$, you need to calculate the probability of each value using the PMF of the binomial distribution. You can then compare the probabilities to determine which value of $x$ has the greatest relative frequency.

Q: What is the value of $x$ that will result in the greatest relative frequency in the binomial distribution?

A: The value of $x$ that will result in the greatest relative frequency in the binomial distribution is the value that has the highest probability. In the case of the binomial distribution, the value of $x$ that will result in the greatest relative frequency is $x = 0$.

Q: Can I use the binomial distribution to model other types of data?

A: Yes, the binomial distribution can be used to model other types of data, such as the number of failures in a fixed number of independent trials, or the number of successes in a fixed number of independent trials with a different probability of success.

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

A: Some common applications of the binomial distribution include:

  • Modeling the number of successes in a fixed number of independent trials
  • Modeling the number of failures in a fixed number of independent trials
  • Modeling the number of successes in a fixed number of independent trials with a different probability of success
  • Modeling the number of failures in a fixed number of independent trials with a different probability of failure

Q: How do I use the binomial distribution in real-world applications?

A: To use the binomial distribution in real-world applications, you need to:

  • Identify the number of trials
  • Identify the probability of success on each trial
  • Use the PMF of the binomial distribution to calculate the probability of each value of $x$
  • Compare the relative frequencies of different values of $x$ to determine which value has the greatest relative frequency

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 that the binomial distribution is a continuous distribution, when it is actually a discrete distribution
  • Failing to account for the number of trials and the probability of success on each trial
  • Failing to use the PMF of the binomial distribution to calculate the probability of each value of $x$
  • Failing to compare the relative frequencies of different values of $x$ to determine which value has the greatest relative frequency.