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Chapter 6: Problem 57

Explain what 鈥渟uccess鈥� means in a binomial probability experiment.

Short Answer

Expert verified
Success means achieving the outcome of interest with a consistent probability in a binomial experiment.

Step by step solution

01

Understanding Binomial Probability

In a binomial probability experiment, there are a fixed number of independent trials, each of which results in one of two outcomes: success or failure. The probability of success is the same in each trial.
02

Defining Success

Success in a binomial experiment is defined as the outcome of interest. For example, if flipping a coin, 'getting heads' can be defined as success.
03

Consistency Across Trials

The probability of success stays constant throughout all the trials in the experiment, denoted as \( p \).
04

Number of Successes

The binomial experiment seeks to determine the probability of achieving a specific number of successes in a given number of trials. The number of successes is denoted as \( k \).

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Key Concepts

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

independent trials
In any binomial probability experiment, independence of trials is a key concept. This means that the outcome of one trial doesn't affect the outcome of any other trial.
For instance, if you roll a die, the result of one roll doesn't influence the result of the next roll.
Each roll is an independent event.
  • Coin flips are another common example: getting heads on one flip has no impact on getting heads on the next flip.
  • This independence is crucial because it ensures the consistency and predictability required in a binomial experiment.
Remember, independence makes it mathematically straightforward to calculate probabilities.
probability of success
The probability of success in a binomial experiment remains constant for each trial.
This probability is usually denoted by the letter \( p \).
Let's consider an example:
  • When flipping a fair coin, the probability of getting a head (success) is 0.5 in each flip.
  • If rolling a 6-sided die, the probability of rolling a 4 (if that's our success) is \( \frac{1}{6} \) each time.
Understanding \( p \) helps to accurately determine the likelihood of achieving a given number of successes in fixed trials.
This constancy ensures that our calculations and predictions are reliable.
fixed number of trials
Binomial experiments always involve a fixed number of trials, which means you know how many repetitions of the experiment will occur beforehand.
This number is usually represented by the letter \( n \).
For example:
  • You might flip a coin 10 times to see how many heads (successes) you get.
  • You might roll a die 20 times to count how many times a 4 appears.
Knowing the exact number of trials is essential as it enables us to use specific mathematical formulas to find probabilities.
A fixed number of trials helps in determining the structure and scope of the experiment, making it manageable and predictable.
outcome of interest
In a binomial probability experiment, the outcome of interest is what we define as a 'success'.
Success isn't limited to positive results; it鈥檚 simply the outcome we are tracking.
Here are some examples:
  • If we are interested in rolling a 4 on a die, then each time a 4 appears, it counts as a success.
  • When flipping a coin and aiming for heads, each head is a success.
This definition allows us to tailor the experiment to our specific goals.
Identifying the outcome of interest is crucial as it directly impacts the calculations and the understanding of the experiment's results.

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Most popular questions from this chapter

In May, \(2000,\) the Gallup Organization reported that \(11 \%\) of adult Americans had a great deal of trust and confidence in the federal government handling domestic issues. Suppose a survey of a random sample of 1100 adult Americans finds that 84 have a great deal of trust and confidence in the federal government handling domestic issues. Would these results be considered unusual? Why?

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=20, p=0.6, x=17 $$

An investment counselor calls with a hot stock tip. He believes that if the economy remains strong, the investment will result in a profit of \(\$ 50,000\). If the economy grows at a moderate pace, the investment will result in a profit of \(\$ 10,000 .\) However, if the economy goes into recession, the investment will result in a loss of \(\$ 50,000\). You contact an economist who believes there is a \(20 \%\) probability the economy will remain strong, a \(70 \%\) probability the economy will grow at a moderate pace, and a \(10 \%\) probability the economy will slip into recession. What is the expected profit from this investment?

The phone calls to a computer software help desk occur at the rate of 2.1 per minute between 3:00 p.m. and 4:00 p.m. Compute the probability that the number of calls between 3:10 p.m. and 3:15 p.m. is (a) exactly eight. Interpret the result. (b) fewer than eight. Interpret the result. (c) at least eight. Interpret the result.

State the criteria for a binomial probability experiment.
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