Question

assume that random guesses are made for 5 multiple - choice questions on a test with 5 choices for each question, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. find the probability of no correct answers. click on the icon to view the binomial probability table. the probability of no correct answers is □ (round to three decimal places as needed.) reference binomial probabilities

Explanation:

Step1: Identify binomial - probability formula

The binomial - probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 5$, $k = 0$, and $p=0.2$.

Step2: Calculate the combination value

When $k = 0$ and $n = 5$, $C(5,0)=\frac{5!}{0!(5 - 0)!}=\frac{5!}{5!}=1$.

Step3: Calculate $(1 - p)^{n - k}$

Since $p = 0.2$, then $1-p=0.8$, and $n - k=5-0 = 5$. So, $(1 - p)^{n - k}=(0.8)^{5}=0.32768$.

Step4: Calculate the probability

$P(X = 0)=C(5,0)\times(0.2)^{0}\times(0.8)^{5}$. Since $(0.2)^{0}=1$ and $C(5,0) = 1$, then $P(X = 0)=1\times1\times0.32768 = 0.328$ (rounded to three decimal places).

Answer:

$0.328$