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(cww), where c denotes a correct answer and w denotes a wrong answer.
p(cww)=\frac{16}{125} (type an exact answer.)
b. beginning with cww, make a complete list of the different possible arrangements of one correct answer and two wrong answers, then find the probability for each entry in the list.
p(cww) - see above
p(wwc)=\frac{16}{125}
p(wcw)=\frac{16}{125}
(type exact answers.)
c. based on the preceding results, what is the probability of getting exactly one correct answer when three guesses are made?
(type an exact answer.)

Explanation:

Step1: Identify probabilities of single - event outcomes

The probability of getting a correct answer ($P(C)$) for a single multiple - choice question with 5 options is $P(C)=\frac{1}{5}$, and the probability of getting a wrong answer ($P(W)$) is $P(W) = 1-\frac{1}{5}=\frac{4}{5}$.

Step2: Calculate probability of getting exactly one correct answer in three guesses

We have three arrangements of one correct and two wrong answers: CWW, WCW, and WWC. Since the probability of each arrangement (using the multiplication rule for independent events) is $P(CWW)=P(WCW)=P(WWC)=\frac{1}{5}\times\frac{4}{5}\times\frac{4}{5}=\frac{16}{125}$.
To find the probability of getting exactly one correct answer in three guesses, we add the probabilities of these mutually - exclusive arrangements.
$P(\text{exactly 1 correct})=P(CWW)+P(WCW)+P(WWC)$.
Since $P(CWW) = P(WCW)=P(WWC)=\frac{16}{125}$, then $P(\text{exactly 1 correct})=\frac{16}{125}+\frac{16}{125}+\frac{16}{125}$.
$P(\text{exactly 1 correct})=\frac{16 + 16+16}{125}=\frac{48}{125}$.

Answer:

$\frac{48}{125}$