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Chapter 6: Problem 20

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=50, p=0.02, x=3 $$

Short Answer

Expert verified
0.0629

Step by step solution

01

Identify the Formula for Binomial Probability

The formula for the binomial probability of getting exactly x successes in n independent trials is given by \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \] where \( n \) is the number of trials, \( x \) is the number of successes, \( p \) is the probability of success on a single trial, and \( \binom{n}{x} \) is the binomial coefficient.
02

Calculate the Binomial Coefficient

Using the binomial coefficient formula: \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \] Substitute \( n = 50 \) and \( x = 3 \): \[ \binom{50}{3} = \frac{50!}{3!(50-3)!} = \frac{50!}{3! \times 47!} \] Calculate the factorials: \[ 50! = 50 \times 49 \times 48 \times 47! \] \[ \binom{50}{3} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 19600 \]
03

Substitute Values into Binomial Probability Formula

Substitute \( \binom{50}{3} = 19600 \), \( p = 0.02 \), and \( n = 50 \): \[ P(X = 3) = 19600 \times (0.02)^3 \times (0.98)^{47} \]
04

Calculate the Probability

Calculate the individual parts: \[ (0.02)^3 = 0.000008 \] and \[ (0.98)^{47} \approx 0.4009 \] Now multiply all together: \[ P(X = 3) = 19600 \times 0.000008 \times 0.4009 \approx 0.0629 \]

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Key Concepts

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

Binomial Coefficient
The binomial coefficient helps us find the number of ways to choose a subset of items from a larger set. This is essential in binomial probability problems. The binomial coefficient is denoted as \(\binom{n}{x}\), which reads as 'n choose x.' It is calculated using the formula: \[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]. Here, \(n\) is the total number of trials, \(x\) is the number of successes we are interested in, and the exclamation mark \( ! \) denotes factorials.
Probability of Success
In a binomial experiment, the probability of success refers to the likelihood of a specific outcome happening on each trial. It is denoted by \(p\). A trial is an individual occurrence in which an event can result in success or failure. For example, if you are flipping a coin, the probability of getting heads (success) is \(p = 0.5\). In our exercise, we have \( p = 0.02 \), meaning each trial has a 2% chance of success.
Factorials
Factorials are mathematical expressions that are very important in binomial probability. They鈥檙e represented by an exclamation point ( \( ! \) ). Calculating a factorial means multiplying a series of descending natural numbers. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). In the binomial coefficient formula \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \], factorials help us figure out how many different ways we can choose \( x \) successes out of \( n \) trials. For instance, in our exercise, we simplify \( 50! \) using the common terms of the numerator and denominator to calculate \( \binom{50}{3} \).

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Most popular questions from this chapter

Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, \(5 \%\) of the patients in the study experienced insomnia as a side effect. A random sample of 20 Clarinex-D users is obtained, and the number of patients who experienced insomnia is recorded. (a) Find the probability that exactly 3 experienced insomnia as a side effect. (b) Find the probability that 3 or fewer experienced insomnia as a side effect. (c) Find the probability that between 1 and 4 patients inclusive, experienced insomnia as a side effect. (d) Would it be unusual to find 4 or more patients who experienced insomnia as a side effect? Why?

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. One hundred randomly selected U.S. parents with at least one child under the age of 18 are surveyed and asked if they have ever spanked their child. The number of parents who have spanked their child is recorded.

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section \(6.1 ;\) (c) compute the mean and standard deviation, using the methods of this section; and \((d)\) draw a graph of the probability distribution and comment on its shape. $$ n=9, p=0.75 $$

In 1898 , Ladislaus von Bortkiewicz published The Law of Small Numbers, in which he demonstrated the power of the Poisson probability law. Before his publication, the law was used exclusively to approximate binomial probabilities. He demonstrated the law's power, using the number of Prussian cavalry soldiers who were kicked to death by their horses. The Prussian army monitored 10 cavalry corps for 20 years and recorded the number \(X\) of annual fatalities because of horse kicks for the 200 observations. The following table shows the data: $$\begin{array}{ll}\hline \text { Number of } & \text { Number of Times } \boldsymbol{x} \\\\\text { Deaths, } \boldsymbol{x} & \text { Deaths Were Observed } \\\\\hline 0 & 109 \\\\\hline 1 & 65 \\\\\hline 2 & 22 \\\\\hline 3 & 3 \\\\\hline 4 & 1 \\\\\hline\end{array}$$ (a) Compute the proportion of years in which there were 0 deaths, 1 death, 2 deaths, 3 deaths, and 4 deaths. (b) From the data in the table, what was the mean number of deaths per year? (c) Use the mean number of deaths per year found in part (b) and the Poisson probability law to determine the theoretical proportion of years that 0 deaths should occur. Repeat this for \(1,2,3,\) and 4 deaths. (d) Compare the observed proportions to the theoretical proportions. Do you think the data can be modeled by the Poisson probability law?

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded.
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