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.
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) =
p(wcw) =
(type exact answers.)

Explanation:

Step1: Calculate probability of a correct answer

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

Step2: Calculate $P(WWC)$

Using the multiplication rule for independent events, $P(WWC)=P(W)\times P(W)\times P(C)$. Substituting the values, we have $P(WWC)=\frac{4}{5}\times\frac{4}{5}\times\frac{1}{5}=\frac{16}{125}$.

Step3: Calculate $P(WCW)$

Again, using the multiplication rule for independent events, $P(WCW)=P(W)\times P(C)\times P(W)=\frac{4}{5}\times\frac{1}{5}\times\frac{4}{5}=\frac{16}{125}$.

Answer:

$P(WWC)=\frac{16}{125}$
$P(WCW)=\frac{16}{125}$