5 Binomial Distribution

Question 1

What is the definition of a Binomial trial?

Answer 1

A trial with exactly two outcomes: success and failure

Question 2

List the conditions for a binomial distribution.

Answer 2

• Exactly two outcomes (success and failure)
• Trials are independent
• Number of trials is fixed
• Probability of success is constant

Question 3

The binomial model is suitable for any random variable that meets which four assumptions?

Answer 3

• Two outcomes
• Independent trials
• Fixed number of trials
• Constant probability of success

Question 4

In a binomial distribution, what does the parameter, n, represent?

Answer 4

The number of trials

Question 5

In a binomial distribution, what does the parameter, p, represent?

Answer 5

The probability of success in a single trial

Question 6

True or false: The probability of success in a binomial distribution can change from trial to trial.

Answer 6

FALSE

The probability of success must remain constant across trials.

Question 7

Fill in the blank: The expectation of a binomial distribution is calculated using the formula _______.

Answer 7

E(X)=np

This formula gives the long-term average of successes in a binomial experiment.

Question 8

Fill in the blank: The variance of a binomial distribution is calculated using the formula _______.

Answer 8

Var(X)=np(1-p)

Variance measures the spread of the distribution around the mean.

Question 9

What is the probability of getting exactly two heads when flipping a fair coin three times?

Answer 9

0.375

Question 10

What is the probability of getting at least 4 heads when flipping a fair coin 10 times?

Answer 10

0.8281

This probability can be calculated using cumulative binomial probabilities.

Question 11

In a binomial distribution, the number of successes is an example of a ________.

Answer 11

discrete random variable

The value of the number of successes varies randomly across trials.

Question 12

What does the notation X ~ B(n, p) signify?

Answer 12

X is a random variable with a binomial distribution

n is the number of trials and p is the probability of success.

Question 13

What is the mean of a binomial distribution with parameters n = 10 and p = 0.5?

Answer 13

5

The mean is calculated as n * p.

Question 14

What is the notatin for a binomial model?

Answer 14

X ~ B(n, p)

n is the number of trials and p is the probability of success.

Question 15

What are the parameters that define a binomial distribution?

Answer 15

n is the number of trials and p is the probability of success.

Question 16

What is the standard deviation of a binomial distribution with parameters n = 10 and p = 0.5?

Answer 16

1.58

The standard deviation is calculated as sqrt(n * p * (1 - p)).

Question 17

What is the probability of getting fewer than 3 heads when flipping a fair coin 5 times?

Answer 17

0.5

This can be calculated using the binomial probability formula.