# A test consists of 5 questions ....

• Mar 30th 2011, 04:59 AM
mathguy80
A test consists of 5 questions ....
Hi All,

A probability problem, not sure whether this is a combination problem or permutation.

A test consists of 5 questions. A correct answer scores 2 marks and an incorrect one, to penalize guessing, scores -1. Assuming that all the questions are answered, find probability of scoring -2.

I figured that only way to score -2 is 1 correct and 4 incorrect, 2 - 4 = -4. All other combinations are incorrect.
Stuck here, Can you guys give me a hint on how to proceed further? Thanks!
• Mar 30th 2011, 06:34 AM
Soroban
Hello, mathguy80!

Quote:

A test consists of 5 questions.
A correct answer scores 2 marks and an incorrect one scores -1.
Assuming that all the questions are answered, find probability of scoring -2.

I figured that only way to score -2 is 1 correct and 4 incorrect: (1)(2) + 4(-1) = -2
. . Right!
Can you guys give me a hint on how to proceed further?

I assume that the student is randomly guessing the answers,
. . and that the probability of guessing a correct answer is $\displaystyle \frac{1}{2}$

To get a score of -2, he must get one Correct and four Incorrect, in some order.

There are 5 choices for the Correct answer.
The probability that answer being correct is $\displaystyle \frac{1}{2}$

He must get the other four equations Incorrect.
The probability of this is: $\displaystyle (\frac{1}{2})^4 \,=\,\frac{1}{16}$

Therefore: .$\displaystyle P(\text{1 Correct, 4 Incorrect}) \;=\;5(\frac{1}{2})(\frac{1}{16}) \:=\:\dfrac{5}{32}$

• Mar 30th 2011, 06:37 AM
Plato
Quote:

Originally Posted by mathguy80
A test consists of 5 questions. A correct answer scores 2 marks and an incorrect one, to penalize guessing, scores -1. Assuming that all the questions are answered, find probability of scoring -2.
I figured that only way to score -2 is 1 correct and 4 incorrect, 2 - 4 = -4.

There are several difficulties with the statement of this question.
It says nothing about the probability of guessing the correct answer to a given question. So let us say that $\displaystyle \mathbf{p}$ is the probability that any given question is answered correctly.
Then the probability of getting exactly one correct is $\displaystyle \dbinom{5}{1}\mathbf{p}(1-\mathbf{p})^4$.
• Mar 30th 2011, 08:36 AM
mathguy80
Thanks @Soroban. Your answer 5/32 checks out. I hadn't made the connection that probability of guessing a correct answer is 1/2.

@Plato, Thanks as well. This is probably an advanced idea, but allowing for variable probability of answering question correctly is very cool. Will keep this in mind for future study.