A quiz consists of 20 multiple-choice questions, each with 4 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 40 %.
I know that we have to do $P(Y/20\geq 0.4)$ and then $P(Y\geq 8)$.
Best Answer
Well on average you would expect them to get $25$% correct with random guesses which is exactly $5$ answers correct so to get $40$% or more correct they would need to get at least $8$ correct.
To get exactly $8$ correct would be $20 \choose 8$ * $.25^8$ * $.75^{12}$ = $.0609.$
To get exactly $9$ correct would be $20 \choose 9$ * $.25^9$ * $.75^{11}$ = $.0271$.
To get exactly $10$ correct would be $20 \choose 10$ * $.25^{10}$ * $.75^{10}$ = $.0099$.
So follow this pattern and sum them up all the way to $20$ correct and that is your answer which should be about $10.18$%.
Any easy way is to use Wolfram Alpha and type in: summation (20 choose n) * .25^n * .75^(20-n) for n=8 to 20