A student takes a multiple-choice exam with 10 questions,...

A student takes a multiple-choice exam with 10 questions, each with four possible selections for the answer. A passing grade is $60 \%$ or better. Suppose that the student was unable to find time to study for the exam and just guesses at each question. Find the probability that the student a. gets at least one question correct. b. passes the exam. c. receives an "A" on the exam (90\% or better). d. How many questions would you expect the student to get correct? e. Obtain the standard deviation of the number of questions that the student gets correct.

A student takes a multiple-choice exam with 10 questions, each with four possible selections for the answer. A passing grade is $60 \%$ or better. Suppose that the student was unable to find time to study for the exam and just guesses at each question. Find the probability that the student
a. gets at least one question correct.
b. passes the exam.
c. receives an "A" on the exam (90\% or better).
d. How many questions would you expect the student to get correct?
e. Obtain the standard deviation of the number of questions that the student gets correct.

Key Concepts

Variance and Standard Deviation

Variance measures the spread of the outcomes in a distribution, showing how much the results differ from the expected value, while the standard deviation is the square root of the variance. In a binomial distribution, these metrics are calculated using the formula np(1-p), offering insight into the variability of the number of successes.

Expected Value

The expected value is the long-run average outcome of a random variable after many repeated trials. For a binomial distribution, it is calculated by multiplying the number of trials by the probability of success, providing an estimate of the number of successes one can expect.

Cumulative Probability

Cumulative probability assesses the probability of achieving a certain number of successes or fewer (or more) across several trials. In the context of exams, it is used to compute the probability of passing or achieving a particular grade by summing probabilities for ranges of correct answers.

Probability of Success

This concept refers to the probability that any individual independent trial results in a successful outcome. In many guessing scenarios, it represents the chance of selecting the correct answer, and it is critical for determining the parameters of the binomial distribution.

Binomial Distribution

The binomial distribution is a discrete probability model that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It is used when there are two outcomes (success or failure) for each trial, which makes it ideal for analyzing guessing on multiple-choice questions.

Complement Rule

The complement rule involves finding the probability that an event occurs by subtracting the probability of the event not occurring from one. This method is particularly useful when calculating the probability of obtaining at least one success, since it can be easier to compute the probability of zero successes.

Transcript

00:01 In this problem, we're given that a student takes a multiple choice exam with 10 questions, each of which has four possible selections for the answer.

00:12 We also told that the passing grade is 60 % or better, and we are to suppose that the student was unable to find time to study for the exam and just guesses at each question.

00:28 So in the first part of the question a, we are finding the probability that the student gets at least one question right.

00:39 So we have to design this distribution and give the success probability p.

00:51 And considering that there are four possible selections and only one answer, there is a 25 % chance that the student is going to get a question right.

01:04 So that means that the success probability is 0 .25.

01:11 And since there are 10 questions to be answered in this exam, n is going to be equal to 10.

01:21 So we need to get the probability, in the first part of the question a, we need to get the probability that x equals or is greater than 1.

01:42 This is the probability that the student gets at least one question right.

01:49 So the probability that the students get at least first question right means we will be adding the probabilities the student got one all the way through until we get to the student got 10 questions.

02:03 In other words, we would have to get one minus the probability that the student did not get any question right.

02:15 So the best thing to do here is to come up with the probability distribution and to do that we need to create a column for x and a column for the probability of the possible outcomes.

02:33 So we have 0, 1, 2, 3 all the way until we get to 10.

02:43 So it's possible that the student would get anything from 0 to 10.

02:49 So the probability that the student gets zero is obtained by the formula.

02:55 You press the equal button and then the binomial distribution, then click on the zero to be the number of questions that were gotten right.

03:09 Out of 10, simple questions, the success probability is 0 .25 and then we check that we are selecting the probability function and we would need to copy that formula all the way down to get the probabilities for the different outcomes and in this case we see now that we want to work out one minus the probability that x equals 0 which is 0 .056 and when we work that out the probability is 0 .9 in the second part of the exercise c, we are finding the probability that the student passes the exam.

04:12 And we have been told that the pass mark is 60 % or better.

04:18 For the student to get 60 % or better, then the number of questions that are correct should be 6 or more...

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