Question

find the indicated probability. a multiple - choice test has 10 questions each of which has 4 possible answers, only one of which is correct. if judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly? 0.750 0.362 0.250 0.506

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 $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Determine values of $n$, $k$, and $p$

Here, $n = 10$ (number of questions), $k = 3$ (number of correct answers), and $p=\frac{1}{4}=0.25$ (probability of getting a single - question correct by guessing), $1 - p = 0.75$.

Step3: Calculate the combination $C(n,k)$

$C(10,3)=\frac{10!}{3!(10 - 3)!}=\frac{10!}{3!7!}=\frac{10\times9\times8}{3\times2\times1}=120$.

Step4: Calculate the probability

$P(X = 3)=C(10,3)\times(0.25)^{3}\times(0.75)^{7}$
$P(X = 3)=120\times0.25^{3}\times0.75^{7}$
$P(X = 3)=120\times0.015625\times0.1334837641$
$P(X = 3)\approx0.250$

Answer:

0.250