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.
p(ccw) = \frac{4}{125} (type an exact answer.)
b. beginning with ccw, make a complete list of the different possible arrangements of two correct answers and one wrong answer, then find the probability for each entry in the list.
p(ccw) - see above
p(cwc) = \frac{4}{125}
p(wcc) = \frac{4}{125}
(type exact answers.)
c. based on the preceding results, what is the probability of getting exactly two correct answers when three guesses are made?
(type an exact answer.)

Explanation:

Step1: Calculate probability of single - event

The probability of getting a correct answer $P(C)=\frac{1}{5}$ and the probability of getting a wrong answer $P(W)=\frac{4}{5}$ since there is 1 correct out of 5 options.

Step2: Calculate $P(CCW)$ using multiplication rule

For independent events, the multiplication rule states that $P(CCW)=P(C)\times P(C)\times P(W)$. Substituting the values, we have $P(CCW)=\frac{1}{5}\times\frac{1}{5}\times\frac{4}{5}=\frac{4}{125}$.

Step3: Calculate $P(CWC)$

$P(CWC)=P(C)\times P(W)\times P(C)=\frac{1}{5}\times\frac{4}{5}\times\frac{1}{5}=\frac{4}{125}$.

Step4: Calculate $P(WCC)$

$P(WCC)=P(W)\times P(C)\times P(C)=\frac{4}{5}\times\frac{1}{5}\times\frac{1}{5}=\frac{4}{125}$.

Step5: Calculate probability of getting exactly 2 correct answers

The probability of getting exactly 2 correct answers out of 3 guesses is the sum of the probabilities of the three arrangements (CCW, CWC, WCC). Since $P(CCW) = P(CWC)=P(WCC)=\frac{4}{125}$, then $P(\text{exactly 2 correct})=\frac{4}{125}+\frac{4}{125}+\frac{4}{125}=\frac{12}{125}$.

Answer:

$\frac{12}{125}$