What is the Probability of Passing a Test by Randomly Guessing?

[kenny1999]

Homework Statement

This is another probability that drives me crazy.

The question is:

A test consists of 6 M.C. questions and each question has 4 options. Students have to answer all questions, 1 mark awarded for each correct answer, no marks deducted for incorrect answers. The passing mark of the test is 4. Suppose a student answers all questions by guessing randomly if he does not prepare for the test.

(a) If Mary forgets to prepare for the test, find the probability that


(iii) Mary passes the test

(b) 40 students take the test and 10 of them do not prepare for the test. Assume students who have prepared for the test can pass the test.

(i) If 2 students are selected from the class at random, find the probability that both of them pass the test.

Homework Equations

The Attempt at a Solution

(a)(iii) is it P(4 answered correct) + P(5 answered correct) + P(6 answered correct)?


(b) (i) This is a big trouble for me. 40 students, 30 students have prepared but 10 students haven't. OK. Because of the assumption given, 30 students should have passed the exam, but how about the remaining 10 students? how can I know how many of them passes the test among the 10 lazy students?

 

[HallsofIvy]

Homework Statement

This is another probability that drives me crazy.

The question is:

A test consists of 6 M.C. questions and each question has 4 options. Students have to answer all questions, 1 mark awarded for each correct answer, no marks deducted for incorrect answers. The passing mark of the test is 4. Suppose a student answers all questions by guessing randomly if he does not prepare for the test.

(a) If Mary forgets to prepare for the test, find the probability that


(iii) Mary passes the test

(b) 40 students take the test and 10 of them do not prepare for the test. Assume students who have prepared for the test can pass the test.

(i) If 2 students are selected from the class at random, find the probability that both of them pass the test.

Homework Equations

The Attempt at a Solution

(a)(iii) is it P(4 answered correct) + P(5 answered correct) + P(6 answered correct)?

Yes. Now what are those probabilitiies?


(b) (i) This is a big trouble for me. 40 students, 30 students have prepared but 10 students haven't. OK. Because of the assumption given, 30 students should have passed the exam, but how about the remaining 10 students? how can I know how many of them passes the test among the 10 lazy students?[/QUOTE]
You don't know but you can calculate the probability. If a student has studied, he has 100% prob of passing. If a student has not studied, the probability of passsing is what you got in (a). What you want is a "weighted average" of those two numbers. A student0picked at random (assumed to mean all 40 students are "equally likely" to be chosen) then the probability he has studied is 30/40=3/4 and the probability he has not is 10/40= 1/4. The probability a student passes is "prob he has studied times prob he passes given that he studied plus prob he has not studied times prob he passes given that he did not study".

 

[tal444]

(a)(iii) Not quite. P(3 answered correct) is also a pass, assuming that 50% is a pass.