Probability question on guessing answers.

Question

There are 3 questions in a multiple choice paper having 4 options (a,b,c,d) each, one of which is correct. If Robert guesses the answers to all these 3 answers as "d", then find the probability of getting one answer correct. It is also noted that the examiner has set the questions and options in such a way that no answer repeats itself, i.e. if any question has an answer as option "a", no other questions will have "a" as an answer.

Answer 1

One important missing statement from this problem is that the test is written so that the correct answer (a,b,c,d) appears randomly, except for the constraint mentioned that no answer is repeated. Why is this important? Well, if the correct test answers to the three questions are: 1-a, 2-b, and 3-c, then if Robert guessed d, he has no chance of getting one correct answer.

Start with question 1 (Q1). The probability that the answer is d is 1/4. The probability for Q2 depends on the result of Q1. If the answer to Q1 was not d (for which there is a .75 probability), then the probability that the correct answer to Q2 is d increases. Why? Because the correct answer to Q2 cannot be the answer that was correct for Q1. For Q2, there are only 3 possible answers. For example, if the correct answer to Q1 was "a", then possible answers for Q2 are only "b", "c" and "d". So, clearly the probability the answer is "d" now increases to 1/3. Similarly, if neither Q1 or Q2 were d, then the probability the answer is "d" for Q3 increased to 1/2.

In equation form:

P[ Robert gets 1 right answer] = P["d" is the answer to Q1] + P["d" is the answer to Q2] + P["d" is the answer to Q3]

P[Robert gets 1 right answer] = 0.25 + P["d" is not the answer to Q1]* 1/3 + P["d" is not the answer to Q1 or Q2] * 1/2

P[Robert gets 1 right answer] = 0.25 + [0.75]* 1/3 + P[0.5]*1/2

P[Robert gets 1 right answer] = 0.25 * 0.25 + 0.25 = 0.75

So Robert has an 0.75 probability of getting one right answer if he guesses "d" to all three questions.

The nice thing about some probability questions is that you can check your answers. This is one of those cases. Create a table with all the possibilities for answers to these 3 questions by creating columns for Q1, Q2 and Q3. For example, the first row would be "a", "b" and "c". The next row could be "a", "b" and "d". Continue on until you have all possibilities. You should find that there are 24 rows in your table (remember - you can't repeat a letter). Of these 24 rows, how many contain a "d". You should find 18. This confirms the probability is 18/24 = 0.75