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Chapter 6: Problem 17

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) success in the \(n\) independent trials of the experiment. \(n=10, p=0.4, x=3\)

Short Answer

Expert verified
The probability is approximately 0.215.

Step by step solution

01

Identify the Binomial Formula

The binomial probability formula is \[ P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} \]where \(n\) is the number of trials, \(p\) is the probability of success on an individual trial, and \(x\) is the number of successful trials.
02

Substitute Given Values

In this problem, \(n=10\), \(p=0.4\), and \(x=3\). Substitute these values into the binomial formula: \[ P(X = 3) = \binom{10}{3} (0.4)^3 (1 - 0.4)^{10 - 3} \].
03

Compute the Binomial Coefficient

The binomial coefficient is calculated as \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \],so \[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \].
04

Calculate the Probability Terms

For \(p^x\), we have \((0.4)^3 = 0.064\).For \((1 - p)^{n - x}\), we have \((0.6)^{10 - 3} = (0.6)^7 = 0.0279936\).
05

Combine All Components

Multiply the binomial coefficient by the probability terms: \[ P(X = 3) = 120 \times 0.064 \times 0.0279936 = 0.21504 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution
In probability and statistics, the binomial distribution is a discrete probability distribution. It describes the number of successes in a fixed number of independent trials. Each trial has two possible outcomes: success or failure.
It's important to know that each trial must be independent, meaning the result of one trial doesn't affect the results of other trials. The formula for the binomial probability is:
\[ P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} \]
This formula helps in finding the probability of having exactly \(x\) successes in \(n\) trials, where \(p\) is the probability of success on any given trial.
Probability of Success
The probability of success, symbolized as \(p\), is the likelihood that an individual trial results in a success. In our example, the probability of success is given as 0.4. This means there's a 40% chance for success in each trial.
It's crucial to distinguish between probability of success \(p\) and probability of failure which is \(1 - p\). If the probability of success \(p\) is 0.4, then the probability of failure is 0.6 because \(1 - 0.4 = 0.6\).
In binomial problems, knowing these probabilities allows you to plug the values into the binomial formula accurately and calculate the desired probabilities.
Binomial Coefficient
The binomial coefficient, also known as \(\binom{n}{x}\), counts the number of ways you can choose \(x\) successes out of \(n\) trials. It is given by the formula:
\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]
Here, \(n!\) denotes the factorial of \(n\), which means multiplying all positive integers from 1 to \(n\). For example, \(10!\) (10 factorial) is equal to \(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
By calculating the binomial coefficient, you determine how many different ways you can achieve exactly \(x\) successes in \(n\) independent trials, which is a crucial step in computing binomial probabilities.
Independent Trials
In a binomial experiment, the trials must be independent. Independence means the outcome of one trial does not influence the outcome of another. Each trial is an isolated event.
This independence assures that the probability \(p\) of success remains constant across all trials. When trials are not independent, the binomial model does not apply, and different statistical methods are required.
An example of independent trials would be flipping a coin multiple times. The result of one flip (heads or tails) does not affect the result of the next flip. Ensuring trials are independent allows for the accurate use of the binomial distribution in calculating probabilities.

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Most popular questions from this chapter

High-Speed Internet According to a report by the Commerce Department in the fall of \(2004,20 \%\) of U.S. households had some type of high-speed Internet connection. (a) Compute the mean and standard deviation of the random variable \(X,\) the number of U.S. households with a high-speed Internet connection in 100 households. (b) Interpret the mean. (c) Would it be unusual to observe 18 U.S. households that have a high-speed Internet connection in 100 households? Why?

On-Time Flights According to American Airlines, its flight 215 from Orlando to Los Angeles is on time \(90 \%\) of the time. Suppose 15 flights are randomly selected and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Find the probability that exactly 14 flights are on time. (c) Find the probability that at least 14 flights are on time. (d) Find the probability that fewer than 14 flights are on time. (e) Find the probability that between 12 and 14 flights, inclusive, are on time.

High-Speed Internet According to a report by the Commerce Department in the fall of \(2004,20 \%\) of U.S. households had some type of high-speed Internet connection. Suppose 20 U.S. households are selected at random and the number of households with high-speed Internet is recorded. (a) Find the probability that exactly 5 households have high-speed Internet. (b) Find the probability that at least 10 households have high-speed Internet. Would this be unusual? (c) Find the probability that fewer than 4 households have high-speed Internet. (d) Find the probability that between 2 and 5 households, inclusive, have high-speed Internet.

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. According to Nielsen Media Research, \(70 \%\) of all U.S. households have cable television. In a small town of 40 households, a random sample of 10 households is asked whether they have cable television. The number of households with cable television is recorded.

Determine the required value of the missing probability to make the distribution a discrete probability distribution. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 3 & 0.4 \\ \hline 4 & ? \\ \hline 5 & 0.1 \\ \hline 6 & 0.2 \\ \hline \end{array}$$
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