Probability of passing this multiple choice exam 
*

*A multiple choice exam has 175 questions. 

*Each question has 4 possible answers. 

*Only 1 answer out of the 4 possible answers is correct. 

*The pass rate for the exam is 70% (123 questions must be answered correctly). 

*We know for a fact that 100 questions were answered correctly. 


Questions: What is the probability of passing the exam, if one were to guess on the remaining 75 questions? That is, pick at random one of the 4 answers for each of the 75 questions. 

The question is essentially what the probability of answering more than 22 questions out of 75 correctly. 
The chance that we will answer all 75 incorrectly is simply $(3/4)^{75}$. The probability that we will answer one particular question correctly and all the rest incorrectly is $(1/4)(3/4)^{74}$. Thus, the probability that we will answer exactly one question incorrectly is $75(1/4)(3/4)^{74}$ since there are 75 possible choices to answer correctly. 
In general, the probability that $p$ questions will be answered correctly will be $${75 \choose p}\left(\frac{1}{4}\right)^p\left(\frac{3}{4}\right)^{75-p}$$ and thus our total probability is
$$P = \sum_{p=23}^{75} {75 \choose p}\left(\frac{1}{4}\right)^p\left(\frac{3}{4}\right)^{75-p} \approx 0.158$$
