Question

a test consists of 10 true/false questions. to pass the test a student must answer at least 6 questions correctly. if a student guesses on each question, what is the probability that the student will pass the test? round to three decimal places.

a. 0.205
b. 0.828
c. 0.377
d. 0.172

Explanation:

Step1: Identify the 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, $n = 10$, $p=\frac{1}{2}$ (since it's a true - false question), and we want $P(X\geq6)=P(X = 6)+P(X = 7)+P(X = 8)+P(X = 9)+P(X = 10)$.

Step2: Calculate $C(n,k)$ for $k = 6$

$C(10,6)=\frac{10!}{6!(10 - 6)!}=\frac{10!}{6!4!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210$.
$P(X = 6)=C(10,6)\times(\frac{1}{2})^{6}\times(1-\frac{1}{2})^{10 - 6}=210\times(\frac{1}{2})^{6}\times(\frac{1}{2})^{4}=210\times(\frac{1}{2})^{10}$.

Step3: Calculate $C(n,k)$ for $k = 7$

$C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10!}{7!3!}=\frac{10\times9\times8}{3\times2\times1}=120$.
$P(X = 7)=C(10,7)\times(\frac{1}{2})^{7}\times(1-\frac{1}{2})^{10 - 7}=120\times(\frac{1}{2})^{7}\times(\frac{1}{2})^{3}=120\times(\frac{1}{2})^{10}$.

Step4: Calculate $C(n,k)$ for $k = 8$

$C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10!}{8!2!}=\frac{10\times9}{2\times1}=45$.
$P(X = 8)=C(10,8)\times(\frac{1}{2})^{8}\times(1-\frac{1}{2})^{10 - 8}=45\times(\frac{1}{2})^{8}\times(\frac{1}{2})^{2}=45\times(\frac{1}{2})^{10}$.

Step5: Calculate $C(n,k)$ for $k = 9$

$C(10,9)=\frac{10!}{9!(10 - 9)!}=\frac{10!}{9!1!}=10$.
$P(X = 9)=C(10,9)\times(\frac{1}{2})^{9}\times(1-\frac{1}{2})^{10 - 9}=10\times(\frac{1}{2})^{9}\times(\frac{1}{2})^{1}=10\times(\frac{1}{2})^{10}$.

Step6: Calculate $C(n,k)$ for $k = 10$

$C(10,10)=\frac{10!}{10!(10 - 10)!}=1$.
$P(X = 10)=C(10,10)\times(\frac{1}{2})^{10}\times(1-\frac{1}{2})^{10 - 10}=1\times(\frac{1}{2})^{10}\times1 = (\frac{1}{2})^{10}$.

Step7: Calculate $P(X\geq6)$

$P(X\geq6)=(210 + 120+45 + 10+1)\times(\frac{1}{2})^{10}=\frac{386}{1024}\approx0.377$.

Answer:

C. 0.377