Determine whether each of the following rvs has a binomial...

Determine whether each of the following rvs has a binomial distribution. If it does, identify the values of the parameters $n$ and $p$ (if possible). (a) $X=$ the number of Eis in 10 rolls of a fair die (b) $X=$ the number of multiple-choice questions a student gets right on a 40 -question test, when each question has four choices and the student is completely guessing (c) $X=$ the same as (b), but half the questions have four choices and the other half have three (d) $X=$ the number of women in a random sample of 8 students, from a class comprising 20 women and 15 men (e) $\quad X=$ the total weight of 15 randomly selected apples (f) $X=$ the number of apples, out of a random sample of $15,$ that weigh more than 150 $\mathrm{g}$

Determine whether each of the following rvs has a binomial distribution. If it does, identify the values of the parameters $n$ and $p$ (if possible).
(a) $X=$ the number of Eis in 10 rolls of a fair die
(b) $X=$ the number of multiple-choice questions a student gets right on a 40 -question test, when each question has four choices and the student is completely guessing
(c) $X=$ the same as (b), but half the questions have four choices and the other half have three
(d) $X=$ the number of women in a random sample of 8 students, from a class comprising 20 women and 15 men
(e) $\quad X=$ the total weight of 15 randomly selected apples
(f) $X=$ the number of apples, out of a random sample of $15,$ that weigh more than 150 $\mathrm{g}$

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent experiments (or trials). Each trial must result in one of two outcomes (success or failure), and the probability of success remains constant across all trials. This distribution is characterized by two parameters: n, the number of trials, and p, the probability of success in a single trial.

Bernoulli Trial

A Bernoulli trial is a basic experiment that results in a binary outcome: success or failure. Each trial in a binomial experiment must be a Bernoulli trial, where the outcome of one trial does not affect the outcome of another, and where the probability of success is the same for each trial.

Fixed Number of Trials

One of the key conditions for a process to be modeled by a binomial distribution is that the number of trials is fixed in advance. This parameter, denoted by n, ensures that the experiment is repeated a set number of times, which is essential to calculating the overall probability distribution of successes.

Independence of Trials

For an experiment to be modeled by the binomial distribution, the trials must be independent – the outcome of one trial does not influence the outcome of any other trial. This assumption of independence is critical because it guarantees that the probability of success remains unchanged from one trial to the next.

Constant Probability of Success

A fundamental assumption of the binomial model is that the probability of success, denoted by p, remains constant across all trials. This means that the chance of achieving a success in any given trial is the same regardless of previous outcomes, which is necessary for the binomial formula to accurately describe the distribution of successes.

Discrete vs. Continuous Random Variables

The binomial distribution applies to discrete random variables, which take on countable values (such as the number of successes). In contrast, continuous random variables, which can take on any value in an interval (like weights or heights), are not modeled by the binomial distribution. This distinction is important when determining whether a given scenario fits the binomial framework.

Transcript

00:01 In this question, a bunch of random experiments are described, and we're asked to identify which ones would have a random variable that is a binomial distribution.

00:09 And if so, we're asked to identify the parameters n and p.

00:15 And so to do this problem, i've just summarized sort of the four criteria from the textbook that are used to determine if it's a binomial random variable or a binomial experiment.

00:30 And just recall that if x is a random, a binomial random variable, we denote it like this, where n is the number of trials, and p is the probability of success in each trial.

00:48 So moving on to part a, we're described x is the number of fours in ten rows of a fair die.

00:55 So we know that there's going to be ten rows of a fair die.

01:00 Success is getting a four.

01:03 Failure is not getting a four.

01:04 And the trials are independent.

01:06 There's no rule of the die affects a different rule of the die.

01:11 So p is constant, which is always one out of six.

01:17 And so yes, x does have a binomial distribution, and we could write it like this.

01:31 For b, it's the number of multiple choice questions a student gets right on a 40 question test when there are four choices for each question, and the student is completely guessing.

01:42 So it's a 40 question test, so there are 40 trials.

01:46 Each trial has success or failure, that's right.

01:48 Student gets it right, it's success.

01:51 The trials are independent.

01:52 That is true because the student is just completely guessing for each question.

01:56 So no answer, getting one right or wrong does not influence getting another question right or wrong.

02:03 And p is constant.

02:04 It is because the student, again, is just guessing.

02:06 So they have a one out of four chance of getting each question right.

02:12 So we can say that x does follow a binomial distribution, and n is equal to 40, and p is equal to 1 over 4.

02:28 Now for questions, c, we're told that it's the same as part b, the same scenario, except that half of the questions have four choices, and the other half have three...

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