Probability of a Multiple Choice test...?

Question

Imagine that you are taking a multiple choice test, 4 possible answer are given for each question. you are to circle the correct answer. suppose there are 20 questions for which you don't know the correct answer. you decide to guess for each of these questions.
1. what is the probability of answering a question correctly ?
2. how many of the 20 question would you expect to answer correctly by guessing ? EXPLAIN
3. in scoring the test, each correct answer is worth 1 point. to discourage guessing, there is a penalty of 1/3 point for each incorrect answer. do you think this is a fair penalty for an incorrect answer ?

Answer 1

1. The probability of answering a question will always be 1 out of the number of answers. So in this case, the probability you will answer a question correctly is 1/4 or 25%. The number of questions doesn't affect the true probability.

2. Based on that, on average, you should answer 5/20 correctly. 20 times 25% is 5.

3. If you have 1/3 penalty for an incorrect answers, by blindly guessing you should break even on average. Since you will answer 1 out of every 4 correctly (scoring 1 point) and answer 3 out of every 4 incorrectly (losing 1/3 of a point each, for a total of 1 full point), you will score 1 minus 1 for a net gain of 0. Here is the catch: If you can eliminate even one answer per question, you will be at an advantage when guessing. Your probability will rise to 1/3. So, you can expect to answer correctly approximately 6 or 7, while only being incorrect 13 or 14 (4-5 points), thus gaining 1-3 points. The more choices you can eliminate, the better your score should be.

On average: At 1/4 you should gain 0 points; at 1/3 you should gain 1-3 points; at 1/2 you should gain 6-7 points (10-3.33).

I think it is a fair penalty (1/3 off) since by blindly guessing you should come out even in the long run. This is fair in that you cannot gain points merely by guessing, but it is not so unfair of a penalty that you should lose points either.

Answer 2

1. Assuming no bias, p= 1/4
2. 20 x 1/4= 5 is expected value.
3. If we answer 5 right and 15 wrong, the scoring system gives us a zero. So it is, on the average, fair.

Answer 3

1. since there are 4 choices, the probablity of answer one question right is 1/4 or 25%

2. since there is 20 questions, you multiply 20 by 25% and you'd get 5 answers.

3. you can earn a total of 20 points. since you might only get 5 answers right, you multiply 5 by 1/3 and subtract it by 20.

Answer 4

Odds don't change. 1 in 4. Maybe more if you're somewhat confident in your answer.

Correct answers? 5 in 20. AKA, 1 in 4.

As for incorrect answers, leaving one blank would be just as bad as guessing incorrectly. Might as well take a stab at it.