##### Question

In: Statistics and Probability

# Multiple-choice questions each have five possible answers (a, b, c, d, e)​, one of which is correct

Multiple-choice questions each have five possible answers (a, b, c, d, e)​, one of which is correct. Assume that you guess the answers to three such questions.

a. Use the multiplication rule to find​ P(CCW), where C denotes a correct answer, and W denotes a wrong answer.

## Solution

##### Step #1

The Counting rule:

Let $$n$$ be the number of events in a sequence, among which the first event $$\left(n_{1}\right)$$ has $$n_{1}$$ possibilities, second event $$\left(n_{2}\right)$$ has $$\left(n_{2}\right)$$ possibilities, the third event $$\left(n_{3}\right)$$ has $$\left(n_{3}\right)$$ possibilities and so on. The total number of possibilities to occur in the sequence is $$n_{1} \times n_{2} \times n_{3} \times \ldots \times n_{r}$$

The formula for counting rule:

The possible number of ways is, $$n !=n(n-1) \ldots 2.1$$. Multiplication principle:

If one of the events happens in $$\mathrm{n}$$ ways and the next event occurs in $$\mathrm{m}$$ ways independently of the previous event. Then the two events can happen in $$n \times m$$ ways. This can be generalized to any number of events.

The objective of the problem is obtained below:

From the information, there are five possible answers for multiple-choice questions that are $$(a, b, c, d, e)$$. $$C$$ denotes the correct answer, and $$\mathrm{W}$$ denotes a wrong answer. The correct answer is probable is $$P(C)=\frac{1}{5}$$, and the probability of the wrong answer is $$P(W)=\frac{4}{5}$$ are used to estimate the $$P(C C W)$$.

The value of $$P(C C W)$$ is obtained below: The required probability value is, $$\begin{array}{c} P(C C W)=\frac{1}{5} \times \frac{1}{5} \times \frac{4}{5} \\ =(0.2 \times 0.2 \times 0.8) \\ =0.032 \end{array}$$

The value of $$P(C C W)$$ is 0.032.