Question

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.

Answer 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.