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Chapter 7: Problem 1

List the conditions required for a binomial experiment.

Short Answer

Expert verified
A binomial experiment requires a fixed number of trials, two outcomes per trial, constant probability of success, and independent trials.

Step by step solution

01

- Fixed Number of Trials

A binomial experiment must have a fixed number of trials, denoted as n. Each trial is an independent and repeatable procedure, meaning the number of trials is determined before the experiment starts and does not change.
02

- Two Possible Outcomes

Each trial in a binomial experiment must have exactly two possible outcomes. These outcomes are often referred to as 'success' and 'failure'.
03

- Constant Probability

The probability of success, denoted by p, must remain constant for each trial. This means that no matter how many trials are conducted, the chance of success in any single trial does not change.
04

- Independent Trials

The outcomes of the trials must be independent of each other. This means that the result of one trial does not affect the outcome of any other trial.

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Key Concepts

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

Fixed Number of Trials
One of the fundamental conditions of a binomial experiment is that it must have a fixed number of trials, denoted by n. This means that before the experiment begins, the total number of times the experiment will be conducted is pre-determined and unchanging. For example, if you are flipping a coin 10 times to see how many heads you get, the number of coin flips (trials) is set at 10.

It鈥檚 important to ensure that each trial is independent and repeatable. This allows for the consistency required to analyze the results statistically. If the number of trials could change mid-experiment, it would be impossible to use the binomial model for analysis.
Two Possible Outcomes
Each trial in a binomial experiment must result in one of exactly two possible outcomes, often labeled as 'success' or 'failure'. This binary nature is essential for applying the binomial probability formula.

For instance, if you are rolling a die, you might define 'success' as rolling a 6 and 'failure' as rolling any other number. These outcomes simplify the analysis by focusing on one desired event and grouping all other outcomes into a single category. This duality makes it easier to compute probabilities and draw conclusions from the experiment.
Constant Probability
Another crucial condition for a binomial experiment is that the probability of success (denoted as p) must remain constant throughout all trials. This constancy implies that the chance of a 'success' in one trial is the same as in any other trial, regardless of previous outcomes.

For example, if you're drawing cards from a deck with replacement, the probability of drawing an Ace remains 1/13 for each draw. This unchanging probability is what allows the binomial distribution to be an effective tool for predicting outcomes over a series of trials.
Independent Trials
The outcomes of each trial in a binomial experiment must be independent of one another. This means that the result of one trial should not influence the result of any other trial. Independence ensures that the probability of success remains constant and that the trials do not interfere with each other.

For example, if you are flipping a coin, the result of one flip (say it lands heads) does not affect the outcome of the next flip. Each flip is its own independent event with its own probability of landing heads or tails.

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Most popular questions from this chapter

State the properties of the standard normal curve.

The heights of 10 -year-old males are normally distributed with mean \(\mu=55.9\) inches and \(\sigma=5.7\) inches. (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of \(10-\) year-old males who are less than 46.5 inches tall. (c) Suppose the area under the normal curve to the left of \(X=46.5\) is \(0.0496 .\) Provide two interpretations of this result.

General Electric manufactures a decorative Crystal Clear 60 -watt light bulb that it advertises will last 1500 hours. Suppose the lifetimes of the light bulbs are approximately normally distributed with a mean of 1550 hours and a standard deviation of 57 hours. (a) What proportion of the light bulbs will last less than the advertised time? (b) What proportion of the light bulbs will last more than 1650 hours? (c) What is the probability that a randomly selected GE Crystal Clear 60 -watt light bulb lasts between 1625 and 1725 hours? (d) What is the probability that a randomly selected GE Crystal Clear 60 -watt light bulb lasts longer than 1400 hours?

Find the indicated probability of the standard normal random variable \(Z\). $$P(-1.20 \leq Z<2.34)$$

Assume the random variable \(X\) is normally distributed with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. $$P(X>35)$$
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