Question

multiple - choice questions each have 5 possible answers, one of which is correct. assume that you guess the answers to 5 such questions. use the multiplication rule to find the probability that the first four guesses are wrong and the fifth is correct. that is, find p(wwwwc), where c denotes a correct answer and w denotes a wrong answer. (round answer to 4 decimal places) p(wwwwc)= what is the probability of getting exactly one correct answer when 5 guesses are made? (round answer to 4 decimal places) p(exactly one correct answer)=

Explanation:

Step1: Calculate probability of wrong answer

The probability of a wrong answer $P(W)$ for a single - multiple - choice question with 5 options is $\frac{4}{5}$ since there are 4 wrong answers out of 5. The probability of a correct answer $P(C)$ is $\frac{1}{5}$.

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

Since the guesses are independent events, we use the multiplication rule for independent events $P(A\cap B)=P(A)\times P(B)$. So $P(WWWWC)=P(W)\times P(W)\times P(W)\times P(W)\times P(C)$. Substituting the values, we get $P(WWWWC)=\frac{4}{5}\times\frac{4}{5}\times\frac{4}{5}\times\frac{4}{5}\times\frac{1}{5}=\frac{4^{4}\times1}{5^{5}}=\frac{256}{3125}= 0.08192\approx0.0819$.

Step3: Calculate probability of exactly one correct answer in 5 guesses

We use the binomial probability formula $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n = 5$ (number of trials/questions), $k = 1$ (number of correct answers), $p=\frac{1}{5}$ (probability of success/correct answer), and $1 - p=\frac{4}{5}$ (probability of failure/wrong answer). The binomial coefficient $C(n,k)=\frac{n!}{k!(n - k)!}$, so $C(5,1)=\frac{5!}{1!(5 - 1)!}=\frac{5!}{1!4!}=5$. Then $P(X = 1)=5\times(\frac{1}{5})^{1}\times(\frac{4}{5})^{4}=5\times\frac{1}{5}\times\frac{256}{625}=\frac{256}{625}=0.4096$.

Answer:

$P(WWWWC)=0.0819$
$P(\text{exactly one correct answer}) = 0.4096$