Question

stan guessed on all 10 questions of a multiple - choice quiz. each question has 4 answer choices. what is the probability that he got at least 2 questions correct? round the answer to the nearest thousandth.
$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$
$_nc_k=\frac{n!}{(n - k)!k!}$
0.211
0.244
0.756
0.944

Explanation:

Step1: Identify parameters

$n = 10$ (number of questions), $p=\frac{1}{4}= 0.25$ (probability of getting a question correct), $1 - p=0.75$ (probability of getting a question wrong).

Step2: Calculate probability of 0 correct answers

$_{n}C_{k}=\frac{n!}{(n - k)!k!}$, for $k = 0$, $_{10}C_{0}=\frac{10!}{(10-0)!0!}=1$.
$P(X = 0)=_{10}C_{0}(0.25)^{0}(0.75)^{10}=1\times1\times(0.75)^{10}\approx0.0563$.

Step3: Calculate probability of 1 correct answer

For $k = 1$, $_{10}C_{1}=\frac{10!}{(10 - 1)!1!}=\frac{10!}{9!1!}=10$.
$P(X = 1)=_{10}C_{1}(0.25)^{1}(0.75)^{9}=10\times0.25\times(0.75)^{9}\approx0.1877$.

Step4: Calculate probability of at least 2 correct

$P(X\geq2)=1 - P(X = 0)-P(X = 1)$.
$P(X\geq2)=1-0.0563 - 0.1877=0.756$.

Answer:

0.756