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Chapter 6: Problem 5

The expected number of successes in a binomial experiment with \(n\) trials and probability of success \(p\) is ______.

Short Answer

Expert verified
The expected number of successes is given by \( n \times p \).

Step by step solution

01

- Understand the Binomial Distribution

A binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. The distribution tracks the number of successes in a given number of trials.
02

- Know the Formula for Expected Value

The formula for the expected number of successes in a binomial experiment is given by the product of the number of trials, n, and the probability of success, p.
03

- Apply the Formula

Multiply the number of trials by the probability of success to find the expected number of successes. This can be expressed as: \[ E(X) = n \times p \]

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Key Concepts

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

Binomial Distribution
The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states. In a binomial experiment, there are exactly two possible outcomes for each trial: a success or a failure.

Here are some essential points to remember:
  • Each trial is independent of the other trials.
  • The probability of success denoted by \( p \), remains constant in every trial.
For example, flipping a coin is a binomial experiment if you label 'heads' as a success and 'tails' as a failure.

When you perform a fixed number of trials \( n \), the binomial distribution helps you find the number of successes.
Expected Value Formula
The expected value (or mean) in the context of a binomial distribution refers to the average number of successes you can expect in a given number of trials.

The formula to calculate the expected value \( E(X) \) in a binomial experiment is straightforward. It is the product of the number of trials \( n \) and the probability of success \( p \). This can be written as:

\[ E(X) = n \times p \]

To help understand this better:
  • If you flip a coin 10 times \( (n = 10) \) with the probability of getting heads \( (p = 0.5) \), the expected value of heads is: \[ E(X) = 10 \times 0.5 = 5 \]
Probability of Success
In a binomial distribution, the probability of success is the probability that one trial will result in a success.

The probability of success is denoted by \( p \). It is essential that \( p \) stays the same from trial to trial in a binomial experiment.

Here are key points to take away:
  • The value of \( p \) is between 0 and 1.
  • The probability of failure is \( 1 - p \).
For instance, if you are rolling a fair six-sided die and you consider rolling a 4 as a success, then the probability of success \( p \) is \[ p = \frac{1}{6} \].

Understanding this concept helps you effectively apply other related formulas like the expected value formula.

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Most popular questions from this chapter

Waiting in Line A Wendy's manager performed a study to determine a probability distribution for the number of people, \(X\), waiting in line during lunch. The results were as follows: $$ \begin{array}{cc|cc} x & \boldsymbol{P}(\boldsymbol{x}) & \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{x}) \\ \hline 0 & 0.011 & 7 & 0.098 \\ \hline 1 & 0.035 & 8 & 0.063 \\ \hline 2 & 0.089 & 9 & 0.035 \\ \hline 3 & 0.150 & 10 & 0.019 \\ \hline 4 & 0.186 & 11 & 0.004 \\ \hline 5 & 0.172 & 12 & 0.006 \\ \hline 6 & 0.132 & & \\ \hline \end{array} $$ (a) Verify that this is a discrete probability distribution. (b) Draw a graph of the probability distribution. Describe the shape of the distribution. (c) Compute and interpret the mean of the random variable \(X\). (d) Compute the standard deviation of the random variable \(X\). (e) What is the probability that eight people are waiting in line for lunch? (f) What is the probability that 10 or more people are waiting in line for lunch? Would this be unusual?

In a recent poll, the Gallup Organization found that \(45 \%\) of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor. (c) Find and interpret the probability that more than 16 of those surveyed feel the state of morals is poor. (d) Find and interpret the probability that 13 or 14 believe the state of morals is poor. (e) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values is poor in the United States? Why?

(a) construct a discrete probability distribution for the random variable \(X\) [Hint: \(\left.P\left(x_{i}\right)=\frac{f_{i}}{N}\right]\), (b) draw a graph of the probability distribution, (c) compute and interpret the mean of the random variable \(X,\) and \((d)\) compute the standard deviation of the random variable \(X\). $$ \begin{array}{cc} x \text { (games played) } & \text { Frequency } \\ \hline 4 & 18 \\ \hline 5 & 18 \\ \hline 6 & 20 \\ \hline 7 & 35 \end{array} $$

Explain how the value of \(p,\) the probability of success, affects the shape of the distribution of a binomial random variable.

(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. $$ n=10, p=0.2 $$
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