Question

An unprepared student must take a 7-question, multiple-choice test that has 5 possible answers per question. If the student can eliminate two of the possible answers on the first three questions, and if she guesses on every question, what is the probability that she will answer at least one question correctly?

Answer 1

Probability of getting 1^st answer correctly = 1/3

(ie, chance of getting correc answer from the 3 multiple choice). This is same for the second and 3^rd answer also.

Probability of getting wrong answer in this case = 2/3

Probability of getting correct answer in case of 4^th case onwards = 1/5

(Because each question has 5 choices and the correct answer is only one out of the 5 choices.)

Probability of getting wrong answer is 4/5

Probability of getting atleast one answer correct.

Case	Case Status	Calculation	Probability
1.	1st one correct & balance all wrong	1/3*2/3*2/3*4/5*4/5*4/5*4/5	1024/16875
2.	2nd one correct & balance all wrong	2/3*1/3*2/3*4/5*4/5*4/5*4/5	1024/16875
3.	3rd one correct & balance all wrong	2/3*2/3*1/3*4/5*4/5*4/5*4/5	1024/16875
4.	4th one correct & balance all wrong	2/3*2/3*2/3*1/5*4/5*4/5*4/5	512/16875
5	5th one correct & balance all wrong	2/3*2/3*2/3*4/5*1/5*4/5*4/5	512/16875
6	6th one correct & balance all wrong	2/3*2/3*2/3*4/5*4/5*1/5*4/5	512/16875
7	7th one correct & balance all wrong	2/3*2/3*2/3*4/5*4/5*4/5*1/5	512/16875
	Total probability	(1024+1024+1024+512+512+512+512)/16875	1024/3375 Ie, 30.34%

Probability of getting at least one answer correct = 30.34%