Probability of one correct answer among 10 questions

Question

I'm not sure how to obtain the answer to problem 6.5.30 R Johnsonbaugh, Discreete Mathematics, 8 ed: "An unprepared student who takes a 10-question true–false quiz and guesses at the answer to every question. What is the probability that the student answers exactly one question correctly?" ChatGPT suggests using the binomial probability formula, but that hasn't been covered in this chapter. How should I think about solving this?

Answer 1

I believe that you are supposed to count the possible number of outcomes for the "rightness/wrongness" of the answers, using the counting techniques from section 6.1, and then count the number of those outcomes that involve exactly 1 right and 9 wrong answers using the concept of "combinations" from section 6.2, and then divide the second by the first to get the probability.

Hint #1. How many possible outcomes of "right" and "wrong" answers are there? I mean like: "all ten wrong", "first is right, next nine are wrong", "first is wrong, second is right, next eight are wrong", etc., etc. Note that this is similar to example 6.1.4 except that instead of choosing 8 bits of "0" or "1", we're choosing 10 answers that are "right" or "wrong".

Hint #2. To count the number of outcomes with exactly one right answer, you want to know the number of ways of choosing one right answer out of 10 total answers. This is the number of 1-combinations of 10 items.

Answer 2

The binomial formula can be used, but isn't necessary in this case. Remember that if every single outcome is equally likely in a finite sample space $S$, any given collection $E$ of events has the following probability of occuring:

\begin{align} P(E)=\frac{|E|}{|S|}. \end{align}

Now, let $E$ denote the subset of possible events in which the student answers exactly one question correctly, and let $S$ denote the set containing every possible combination of quiz answers.

Answer 3

Since you haven't studied the binomial distribution, one sequence of Right/Wrong answers she could have given is $WWWRWWWWWW$ with $Pr = 0.5^{10}$

But the right answer could be any of the ten, so multiply by 10 to get the answer asked for

Answer 4

You already have some correct answers. Here is an approach which is a little different.

There are $2^{10}$ possible sequences of right / wrong answers, all of which we assume are equally likely. In how many of these sequences is exactly one answer right? There are $10$ ways to pick the associated question, and then the answers to the remaining questions must all be wrong. So the probability of getting exactly one right answer is $$\frac{10}{2^{10}}$$