5.3 CPS Oswego High School

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Questions and Answers

Which of the following is NOT a requirement for a procedure to represent a binomial distribution?

The trials must be independent.

The probability of a success must vary across trials. (correct)

The procedure has a fixed number of trials.

Each trial must result in one of two outcomes.

If a procedure has a sample size that represents 10% of the total population, how should the trials be treated?

As independent events.

As continuous events.

As dependent events. (correct)

As mutually exclusive events.

In a binomial probability distribution, what does the symbol 'n' represent?

The number of successes in trials.

The total number of trials conducted. (correct)

The probability of failure.

The probability of success.

Which of the following best describes the concept of 'success' in a binomial probability distribution?

An arbitrary outcome denoted as S that can be negative in context. Signup and view all the answers

What does the letter 'p' signify in the notation of binomial probability distributions?

The probability of success in one trial. Signup and view all the answers

In a binomial distribution, if the probability of failure is denoted by 'q', how is it mathematically expressed in terms of 'p'?

q = 1 - p Signup and view all the answers

Why must the trials in a binomial distribution be independent?

To make sure that prior trials do not affect the current trial's outcome. Signup and view all the answers

In determining the probability of an event in a binomial distribution, what might need to be considered to establish the value of 'p'?

The complement of the given probability. Signup and view all the answers

What does P(x) represent in a Binomial Probability Distribution?

The probability of getting exactly x successes Signup and view all the answers

Which of the following values represents the probability of failure in a Binomial experiment where the probability of success is 0.75?

0.25 Signup and view all the answers

How can the probability of getting less than 6 heads in 20 coin tosses be identified using Binomial functions?

binomcdf(20, 0.5, 5) Signup and view all the answers

In the given experiment with 5 offspring peas, what is the probability of getting exactly 3 green pod offspring?

0.2637 Signup and view all the answers

Which is NOT a condition when using the Binomial Probability Distribution?

The probability of success must change with each trial Signup and view all the answers

When sampling without replacement, under what condition can the events be considered independent?

If n < 0.05N Signup and view all the answers

What is the correct sequence to input values into a TI-83 when using the binompdf function?

n, p, x Signup and view all the answers

The cumulative distribution function (CDF) for a binomial probability distribution is represented by which notation?

binomcdf Signup and view all the answers

What is the probability of getting at least 14 heads when tossing a coin 32 times?

0.811 Signup and view all the answers

In a class of 25, what is the probability that exactly 1 person is allergic to bee stings?

0.196 Signup and view all the answers

Calculate the probability that fewer than 3 students in a class of 9 receive financial aid, given that 53% of students receive financial aid.

0.064 Signup and view all the answers

When finding the probability that 4 or more people in a class of 25 are allergic to bee stings, which method should be used?

Use the complement method: 1 – P (x ≤ 3). Signup and view all the answers

What conditions must be satisfied for a procedure to qualify as a binomial distribution?

Independent trials, a fixed number of trials, and the same probability for each trial. Signup and view all the answers

Given the scenario, what represents the parameters for a binomial distribution where 571 subjects out of 863 said 'yes'?

n = 863, x = 571, p = 0.5, q = 0.5 Signup and view all the answers

If the probability of receiving financial aid is 53%, what is the probability of exactly 5 students receiving it in a group of 9?

0.257 Signup and view all the answers

What is the probability that at least 14 out of 32 coin tosses yield heads?

0.811 Signup and view all the answers

Is the scenario of treating 152 couples with YSORT gender selection method a binomial distribution?

No, subjects are not independent. Signup and view all the answers

In the scenario involving 20 Senators, which factor contributes to the situation not being a binomial distribution?

Senators are chosen without replacement. Signup and view all the answers

Given the scenario where n = 12, x = 10, and p = 3/4, what is the probability of exactly 10 successes?

0.2323 Signup and view all the answers

What does the 5% guideline refer to in the context of probability sampling?

The sample size must not exceed 5% of the population. Signup and view all the answers

What is one condition that must be met for a procedure to be classified as having a binomial distribution?

Trials must be independent. Signup and view all the answers

What are the values of n, x, p, and q in the context of the binomial distribution if n = 20, x = 4, and p = 0.15?

n=20, x=4, p=0.15, q=0.85 Signup and view all the answers

What is a characteristic of the binomial probability formula?

It calculates the probability of multiple successes. Signup and view all the answers

Why can't the 5% guideline be applied in the Senate example?

The sample size was too large compared to the population. Signup and view all the answers

What is the probability that at least one of the 5 donors has Group O blood?

0.949672 Signup and view all the answers

If the probability of getting heads in 20 coin tosses is studied, what is the probability of getting exactly 9 heads?

0.1602 Signup and view all the answers

In a class of 25 people with a 1% allergy rate, what is the probability that exactly 1 person is allergic to bee stings?

0.0041 Signup and view all the answers

What is the probability that at least 7 out of 10 consumers recognize the Mrs.Fields brand?

0.9872 Signup and view all the answers

What is the probability of getting less than 6 heads in 20 coin tosses?

0.0207 Signup and view all the answers

In a random group of 9 students, what is the probability that exactly 5 of them receive financial aid?

0.2571 Signup and view all the answers

What is the probability of obtaining at least 3 out of 5 people who have heard of McDonald's?

0.9988 Signup and view all the answers

What is the probability that 4 or more individuals in a class of 25 are allergic to bee stings?

0.0001 Signup and view all the answers

Study Notes

Binomial Probability Distributions

Binomial probability distributions deal with circumstances where outcomes fall into two categories (e.g., pass/fail, success/failure).
A binomial distribution results from a procedure with these characteristics:
- A fixed number of trials.
- Independent trials (one outcome doesn't affect others). A sample less than 5% of the population can be treated as independent.
- Each trial has two outcomes (success or failure).
- The probability of success remains constant for each trial.

Notation

S and F represent success and failure, respectively.
p represents the probability of success.
q represents the probability of failure (q = 1 - p).
n represents the fixed number of trials.
x represents a specific number of successes in n trials (0 ≤ x ≤ n).
P(x) represents the probability of getting exactly x successes in n trials.

Formula for Calculating Probability

P(x) = n! / ((n-x)! * x!) * p^x * q^(n-x)
- n! = n factorial
- p = probability of success
- q = probability of failure (q = 1 - p)

Calculating Probability Using a Calculator

Use the binompdf function on a graphing calculator to calculate a specific probability of exactly x successes in n trials.
Use the binomcdf function to calculate the cumulative probability of x or fewer successes.
For finding the probability of at least x successes, subtract the probability of less than x successes from 1.

Important Considerations

Be sure x and p represent the same category of success.
When dealing with no replacement, consider events to be independent if n < 0.05N (where N is the population size) .

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Description

This quiz focuses on binomial probability distributions, which describe scenarios with two possible outcomes, such as success and failure. It covers key concepts including notation, fixed trials, and how to calculate the probability of successes using specific formulas. Test your understanding of these statistical principles!