# Probability of same answer choice 4-in-a-row on exams

#### muabbas

I would greatly appreciate everyone's help with the following (related) probability questions. I believe they involve something called binomial probability but it's been several years since I've taken any math course and don't believe I ever took any math that dealt with these kinds of questions:

On an multiple-choice exam's answer key, what are the chances of a particular answer choice appearing 4 times in a row, assuming there are 5 answer choices (A, B, C, D, & E) and 25 questions on the exam? How many 4-in-a-rows would one expect over the course of 256 25-question exams?

What are the chances of having 2 4-in-a-row occurrences in the same 25-question exam? The letter doesn't have to be the same for each occurrence. How many of these would one expect over the course of 256 25-question exams?

What are the chances of 9 4-in-a-row occurrences over the course of 256 25-question exams? Again, the letter doesn't have to be the same for each occurrences.

What are the chances of a particular 5-answer sequence (such as "ABCDE") occurring 3 times over the course of 256 25-question exams? How many ABCDE sequences would one expect over the course of 256 25-question exams?

By the way, I want to assume that the answer choices are randomly generated (no human bias in avoiding certain sequences).

Thank you very much!

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#### shenanigans87

I would greatly appreciate everyone's help with the following (related) probability questions. I believe they involve something called binomial probability but it's been several years since I've taken any math course and don't believe I ever took any math that dealt with these kinds of questions:

On an multiple-choice exam's answer key, what are the chances of a particular answer choice appearing 4 times in a row, assuming there are 5 answer choices (A, B, C, D, & E) and 25 questions on the exam?
The chance of four questions in a row having the same answer, say "A", is $$\displaystyle (\frac{1}{5})^4\;=\;\frac{1}{625}$$

Consider that if there are 25 questions, you have 22 opportunities to get 4 in a row. Think of it like this

1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
. . .
21 22 23 24
22 23 24 25

So the chances of getting one set of 4-in-a-row answers on a 25 question exam is

$$\displaystyle (\frac{624}{625})^{21}*(\frac{1}{625})^1\;=\;0.1547$$%

• downthesun01