6.3: The Binomial Distribution

Question 1

What is the binomial distribution in the context of probability?

Answer 1

A binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Question 2

What constitutes a ‘trial’ in the binomial distribution context?

Answer 2

A trial is one instance of an event that can result in a success or a failure. In the binomial distribution, all trials are identical and independent of each other.

Question 3

In a binomial distribution, what do the terms ‘success’ and ‘failure’ signify?

Answer 3

‘Success’ refers to the outcome of interest or the event we are counting, while ‘failure’ refers to all other outcomes.

Question 4

How do you represent the probability of success (P(S)) and failure (P(F)) in a binomial distribution?

Answer 4

P(S) is the probability of a success on a single trial, and P(F) is the probability of a failure on a single trial, where P(F) = 1 - P(S).

Question 5

What is the assumption about the trials in a binomial experiment?

Answer 5

The trials are independent, meaning the outcome of one trial does not affect the outcome of another.

Question 6

How is the general probability of x successes in n trials calculated in a binomial distribution?

Answer 6

The probability is calculated as

p(x) = (n! / (x!(n - x)!)) * p^x * q^(n - x),

where p is the probability of success on a single trial, q is the probability of failure on a single trial, n is the number of trials, and x is the number of successes.

Question 7

What does the binomial coefficient (n choose x) indicate?

Answer 7

The binomial coefficient (n choose x) indicates the number of ways to arrange x successes among n trials in a binomial experiment.

Question 8

What is a binomial random variable?

Answer 8

A binomial random variable x represents the total number of successes in n trials of a binomial experiment.

Question 9

How do you find the probability of obtaining exactly x successes in n binomial trials?

Answer 9

The probability is found by multiplying the number of ways to arrange x successes among n trials by the probability of achieving x successes and (n-x) failures, which is given by p^x * q^(n-x).

Question 10

How do you calculate the probability of exactly three successes in five trials for a binomial distribution?

Answer 10

The probability is calculated using the formula p(3) = 5! / [3!(5-3)!] * (p)^3 * (q)^(5-3), where p is the probability of success on a single trial, and q is the probability of failure (q = 1-p).

Question 11

What does the term ‘success’ in a binomial experiment refer to?

Answer 11

In the context of a binomial experiment, ‘success’ does not necessarily mean a positive or desirable outcome; it simply refers to one of the outcomes that we are interested in tracking, whether positive or negative.

Question 12

How can binomial tables be used to find binomial probabilities?

Answer 12

Binomial tables provide the probability of x successes in n trials for different values of p (probability of success).

To use the table, locate the desired value of p and n, and find the intersection that corresponds to the number of successes.

Question 13

How do you interpret probabilities from a binomial probability table?

Answer 13

Probabilities from a binomial table can be interpreted as the likelihood of a certain number of successes in a set number of trials, given a specific probability of success.

Question 14

What is the ‘rare event approach’ in statistical inference?

Answer 14

The rare event approach suggests that if the probability of an observed sample result under a given assumption is small, we have strong evidence that the assumption is false.

Question 15

What is the binomial probability of 5 successes in 8 trials for a success probability of 0.95?

Answer 15

The binomial probability

p(5) = 8! / [5!(8 - 5)!] * (0.95)^5 * (0.05)^(8-5) = .0054.

Question 16

How do binomial tables simplify finding probabilities for binomial distributions?

Answer 16

Binomial tables list probabilities for different numbers of successes (x) in a set number of trials (n) across various success probabilities (p), eliminating the need for manual calculation.

Question 17

What does the probability p(x ≤ 5) = .0058 indicate in the context of 8 television sets lasting five years without repair?

Answer 17

It indicates that there is only a 0.58% chance of five or fewer out of eight televisions lasting five years without repair if the true probability of no repair is 0.95, suggesting strong evidence against the manufacturer’s claim.

Question 18

What are the parameters of a binomial distribution?

Answer 18

The parameters of a binomial distribution are the number of trials n, and the probability of success on each trial p.

Question 19

How can you calculate the mean, variance, and standard deviation of a binomial random variable?

Answer 19

For a binomial random variable with parameters n and p, the mean is μx = np, the variance is σx² = npq, and the standard deviation is σx = √(npq), where q = 1 - p.