Aa student takes a $10$-question true/false exam and guesses. What is the probability that the student answers every question incorrectly? 
Suppose a student takes a $10$-question true/false exam and guesses at every question.
  What is the probability that...
  a) the student answers every question incorrectly?
  b) the students answers at least one question correctly?
  c) the student answers exactly 5 questions correctly?   

So far I have solved a. I got $(1/2)^{10}$.
For b I'm not sure what to do.
For c would the answer be $(1/2)^5$? 
 A: a) Is fine.
b) Notice that the complementary event of $A =$"at least one right" is $\bar A =$"all wrong". Hence
$$P(A) = 1-P(\bar A) = 1-\left(\frac{1}{2}\right)^{10}.$$
c) No. You have to count all the spots where you got them right. Notice that there are $\binom{10}{5}$ ways to choose the questions that are correct. We also know that $5$ are wrong (with chance $1/2$ each time) and $5$ are right (with chance $1/2$ each time). Hence
$$P(\text{Exactly 5 right}) = \binom{10}{5}\left(\frac12\right)^5\left(\frac12\right)^5.$$
The number of correct answers in $10$ tries follows a binomial distribution.
A: Given that the probability of answering a question correctly is $\dfrac12$:


a) the student answers every question incorrectly?

$$\left(1-\frac12\right)^{10}$$


b) the students answers at least one question correctly?

$$1-\left(1-\frac12\right)^{10}$$


c) the student answers exactly 5 questions correctly?

$$\binom{10}{5}\cdot\left(\frac12\right)^{5}\cdot\left(1-\frac12\right)^{10-5}$$
A: Let $X$ be a random variable that maps the number of correct answers guessed to its probability. $X$ is discrete and thus $X \sim Bionomial(n=10, p = \frac{1}{2})$. Just plug in the corresponding $P(X = a)$ and calculate via. the binomial distribution equation to compute your desired answer. For part b it is $P(X \geq 1) = 1 - P(X =0)$
A: For b it will be $$\sum\frac{{10\choose i}}{2^{10}}$$ where $ i \in [1,10]$ as its atleast 1 correct . For c it will be any $5$ from $10$ so it will be $$\frac{{10\choose 5}}{2^{10}}$$
