Question

When taking a test with m closed answers, a student knows the correct answer with probability p, otherwise he chooses one of the possible answers at random. What is the probability that the student knows the correct answer given that he answered the question correctly.

182

likes
912 views

Answer to a math question When taking a test with m closed answers, a student knows the correct answer with probability p, otherwise he chooses one of the possible answers at random. What is the probability that the student knows the correct answer given that he answered the question correctly.

Expert avatar
Frederik
4.6
64 Answers
To solve this problem, we can use Bayes' theorem.

Let's denote the following events:
A: The event that the student knows the correct answer.
B: The event that the student answered the question correctly.

We are asked to find P(A|B), the probability that the student knows the correct answer given that he answered the question correctly.

According to Bayes' theorem, we have:

P(A|B) = \frac{{P(B|A) \cdot P(A)}}{{P(B)}}

We can calculate each of these probabilities step-by-step:

1. P(A) is the probability that the student knows the correct answer. This is given as p.
2. P(B|A) is the probability that the student answered the question correctly given that he knows the correct answer. This is equal to 1 since we are assuming that the student knows the correct answer.
3. P(B) is the total probability that the student answered the question correctly.

To calculate P(B), we need to consider two cases:
a) The student knows the correct answer, which happens with probability p.
b) The student does not know the correct answer, which happens with probability 1 - p. In this case, the probability of answering correctly by randomly choosing one of the possible answers is 1/m.

Therefore, we have:

P(B) = P(A) \cdot 1 + (1 - P(A)) \cdot \frac{1}{m} = p + \frac{1 - p}{m}

Now we can substitute these values back into Bayes' theorem to find P(A|B):

P(A|B) = \frac{{1 \cdot p}}{{p + \frac{1 - p}{m}}} = \frac{{p \cdot m}}{{pm + 1 - p}}

Answer: The probability that the student knows the correct answer given that he answered the question correctly is \frac{{p \cdot m}}{{pm + 1 - p}}

Frequently asked questions (FAQs)
What is the intersection point of the exponential functions f(x) = 10^x and f(x) = e^x?
+
Question: What is the value of x in a right triangle if one acute angle measures 30 degrees and the hypotenuse is 8 units long?
+
Math Question: What is the limit of (3x^2 - 5x + 2) as x approaches 2?
+
New questions in Mathematics
calculate the derivative by the limit definition: f(x) = 6x^3 + 2
Add. 7/w²+18w+81 + 1/w²-81
Eight acts are scheduled to perform in a variety show how many different ways are there to schedule their appearances show your work
Consider the relation R defined on the set of positive integers as (x,y) ∈ R if x divides y. Choose all the true statements. R is reflexive. R is symmetric. R is antisymmetric. R is transitive. R is a partial order. R is a total order. R is an equivalence relation.
STUDENTS IN A CLASS LEARN ONLY ONE FOREIGN LANGUAGE. two-sevenths of the students learn German, half of the students learn Spanish, and the remaining six students learn Italian. what is the number of students in this class? detail your reasoning carefully.
A car that starts from rest moves for 11 min, reaching a speed of 135 km/h, calculate the acceleration it had
The profit G of the company CHUNCHES SA is given by G(x) = 3×(40 – ×), where × is the quantity of items sold. Find the maximum profit.
[(36,000,000)(0.000003)^2]divided(0.00000006)
The beta of a company is 1.51 while its financial leverage is 27%. What is then its unlevered beta if the corporate tax rate is 40%? (4 decimal places)
Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll? Draw the diagram
In a grocery store, when you take out 3 peppers and 4 carrots, there are 26 peppers and 46 carrots left. How many peppers and carrots were there initially?
If a two-branch parallel current divider network, if the resistance of one branch is doubled while keeping all other factors constant, what happens to the current flow through that branch and the other branch? Select one: a. The current through the doubled resistance branch remains unchanged, and the current through the other branch decreases. b. The current through the doubled resistance branch decreases, and the current through the other branch remains unchanged. c. The current through the doubled resistance branch increases, and the current through the other branch remains unchanged. d. The current through both branches remain unchanged.
0.1x8.2
(2m+3)(4m+3)=0
The grading on a $159,775 house comes to $3974.75. What percent of the total cost is this? (Express your answer to the nearest hundredth percent.)
Write the detailed definition of a supply chain/logistics related maximization problem with 8 variables and 6 constraints. Each constraint should have at least 6 variables. Each constraint should have At least 5 variables will have a value greater than zero in the resulting solution. Variables may have decimal values. Type of equations is less than equal. Numbers and types of variables and constraints are important and strict. Model the problem and verify that is feasible, bounded and have at least 5 variables are nonzero.
Write the inequality in the form of a<x<b. |x| < c^2
write in set builder notation { 1,3,9,27,81,243,...}
Write decimal as the fraction 81/125 simplified
Let A denote the set of all people who were alive in 2010. Let B denote the set of all real numbers. Let f assign, to each person in A, their weight during the year 2010. Is f a function? Explain in complete sentences.