Whats the chance of getting 25/40 multiple choice questions right.

On an exam with 40 questions, each question having 4 different options with only one right answer. If you need 25/40 questions right to pass, whats the odds of you getting 25 right by guessing?

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1 Answer

The odds of answering exactly 25 questions correectly is 
\[{40 \choose 25}(\frac{1}{4})^{25}(\frac{3}{4})^{15}=0.00000047744016\approx 0\%\]


You will pass if you answer at least $25$ questions correcxtly. So the probability is 
\[{40 \choose 25}(\frac{1}{4})^{25}(\frac{3}{4})^{15}+{40 \choose 26}(\frac{1}{4})^{26}(\frac{3}{4})^{14}+\dots +{40 \choose 40}(\frac{1}{4})^{40}(\frac{3}{4})^0\]
\[=\sum_{n=25}^{40}{40 \choose 25}(\frac{1}{4})^{n}(\frac{3}{4})^{40-n}=0.00000058796594= 0\%.\]
Indeed ${40 \choose 25}(\frac{1}{4})^{n}(\frac{3}{4})^{40-n}$ is the probability that you answer exactly $n$ questions correctly. In order to pass $n$ could be any number between $25$ and $40$ and that is why we are adding the probabilities. You can use this website to calculate probabilities:

https://www.gigacalculator.com/calculators/binomial-probability-calculator.php

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