Probability of Multiple Choice first attempt and second attempt A multiple choice question has 5 available options, only 1 of which is correct. Students are allowed 2 attempts at the answer. A student who does not know the answer decides to guess at random, as follows:
On the first attempt, he guesses at random among the 5 options. If his guess is right, he stops. If his guess is wrong, then on the second attempt he guesses at random from among the 4 remaining options. Find the chance that the student gets the right answer at his first attempt? Then, find the chance the student has to make two attempts and gets the right answer the second time?Find the chance that the student gets the right answer?
$P(k)=nk\times p^k\times({1−p})^n−k$.
$P($First attempt to get right answer$)=(5C_1)\times \frac{1}{5} \times (\frac{2}{4})^4=?$
$P($Second attempt to get right answer$)=(5C_2)\times \frac{1}{5}\times (\frac{2}{4})^3=?$
$P($The student gets it right$)=(5C_1)\times \frac{1}{5} \times (\frac{2}{4})^4=?$
 A: The probability that the student gets the right answer the first time is $\frac{1}{5}$. For there are $5$ possible first guesses, only $1$ of which is right. 
There is a long (well, not that long) way of doing the second problem. Let's do it quickly. Imagine that the student has prepared a list of guesses in advance. The probability that the right answer is in first position among these guesses is exactly the same as the probability it is in the second position, the third, the fourth, the fifth: they are all $\frac{1}{5}$.  
So the probability that she is  wrong on the first and right on the second is $\frac{1}{5}$. If more guessing were allowed, the probability she is wrong on the first two guesses, and right on the third, is again $\frac{1}{5}$. And so on. 
A: First try to find the sample space $S$ for the question. There are five equally likely choices, so $S=\{c_1,\cdots, c_5\}$ and the event $E \subset S$ is choosing the correct answer, and there is only one correct answer i.e. $|E|=1.$ Therefore the probability is $\frac{|E|}{|S|}=\frac 15.$
Do the same to determine the sample space for the second question and find the relevant event. What is the answer, then?
