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Question

# A test has three parts. Part A consists of eight true-false questions, Part B consists of five multiple choice questions with five choices each, and Part C requires you to match five questions with five different answers one-to-one. Assuming that you make random guesses in filling out your answer sheet, what is the probability that you will earn 100% on the test? (Leave your answer as a formula.)

Solution

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Recall that the probability of an event $E$ in case all the outcomes are equally likely is

$$$P(E) = \dfrac{n(E)}{n(S)}$$$

where $S$ is the sample space.

The sample space $S$ consists of all possible answer sheets to the test. Let's find the cardinality of $S$ using the following decision algorithm:

$\textbf{Step 1}$. Solve part A. There are 8 true-false questions. Each of them can be solved in 2 ways (either true or false). According to multiplication principle, there are $2^8$ ways to solve part A.

$\textbf{Step 2}$. Solve part B. There are 5 questions with 5 choices each. Each of them can be solved in 5 ways. According to multiplication principle, there are $5^5$ ways to solve part B.

$\textbf{Step 3}$. Solve part C. There are 5 question that need to be matched with 5 answers. There are $5!$ possible ways to arrange 5 answers to fit the questions.

Using multiplication principle, we get

$n(S) = 2^8\times 5^5 \times 5!$

Let $E$ be the event that all the answers are correct. Obviously there is only one outcome in $E$.

$n(E) = 1$

Using formula (1), we get $P(E) = \dfrac{1}{2^8\times 5^5 \times 5!}$

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