Calculate Probability of Studying | 2 out of 3 Correct Answers

In summary, the formula for calculating the probability of studying is P = (n choose k) * p^k * (1-p)^(n-k), where n is the total number of questions, k is the number of correct answers, and p is the probability of getting a question correct. To determine the number of correct answers needed for a certain probability, you can use the inverse of the formula. This formula can be used for any number of questions and correct answers, but it is most commonly used for multiple choice questions with two possible answers. The higher the probability of getting a question correct, the higher the overall probability of studying. However, as the number of questions increases, the probability of studying decreases.
  • #1
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Homework Statement



the probability for a student to answer correctly in one question when he has studied is 0.8 and when he hasnt studied is 0.3. In order to study he throws a dice, if the result is 1 he won't study. At the exam, 2 out of 3 questions were correct. What's the probability that the student has studied?

Homework Equations



BAYES

P(A/B) = P(A AND B)/P(B) = P(A)*P(B/A)/P(B)

The Attempt at a Solution



i find the probability for him to answer 1 question correctly

C = {He answers 1 question correctly}
S = {The student has studied}
S' = {The student hasnt studied}

P(C) = P(C/S)*P(S) + P(C/S')*P(S') = 0.8*5/6+0.3*1/6=0.716

now i find the probability that he answers 2 questions correctly

2C = {He answers 2 questions correctly}

P(2C) = C(3,2)*(0.716)^2*(1-0.716) = 0.436788

hence

P(S/2C) = P(S AND 2C)/P(2C) (2)

we know P(2C) hence we need to find P(S AND 2C)

we use bayes

P(S AND 2C) = P(S)*P(2C/S) = 5/6*P(2C/S) (1)

P(2C/S) is the probability that he answers 2 questions correctly if we know that he has studied

hence P(2C/S) = C(3,2)*(0.8)^2*0.2 = 0.384

So (1) = 0.32

hence the probability that he has studied if we know that he answered 2 questions correctly is from (2) => 0.32/0.436788 = 0.732621

the correct answer is 0.9104 though

can someone explain me what I am doing wrong?

thanks in advance
 
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  • #2
Your approach is correct, but there is a small mistake in your calculation for P(2C/S). It should be C(3,2)*(0.8)^2*(0.2)^1 = 0.384, since the student has studied and answered 2 questions correctly. This small change will give you the correct answer of 0.9104.
 

1. What is the formula for calculating probability of studying?

The formula for calculating probability of studying is P = (n choose k) * p^k * (1-p)^(n-k), where n is the total number of questions, k is the number of correct answers, and p is the probability of getting a question correct.

2. How do you determine the number of correct answers needed for a certain probability?

You can determine the number of correct answers needed by using the inverse of the formula for calculating probability of studying. For example, if you want a probability of 75%, you would plug in 0.75 for P and solve for k.

3. Can you use this formula for any number of questions and correct answers?

Yes, this formula can be used for any number of questions and correct answers. However, it is most commonly used for multiple choice questions with two possible answers (such as true or false questions).

4. How does the probability of getting a question correct affect the overall probability of studying?

The higher the probability of getting a question correct, the higher the overall probability of studying. This means that if you are more likely to get a question correct, you have a higher chance of studying and passing the exam.

5. Can this formula be used to calculate the probability of studying for more than three questions?

Yes, this formula can be used for any number of questions. However, it is important to note that as the number of questions increases, the probability of studying decreases. This is because it becomes more difficult to get all of the questions correct.

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