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If I have a test with one question which is true or false, the chances of getting $100\%$ in the exam is $50\%$ as I can select a right or wrong answer.

But what is the chance (exact percentage figure) of getting $100\%$ in a test where there are $50$ questions, each having true or false?


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up vote 3 down vote accepted

Assuming that you’re guessing on each question, so that the probability of getting any given question right is $\frac12$, the probability of guessing right on all $50$ questions is $\left(\frac12\right)^{50}$. This is a little more than $8.88\times 10^{-16}$.

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So the exact percentage is 0.0000000000000888178%? – user1394925 Mar 30 '13 at 23:18
Can I ask what the dormula you used is called btw as I havn't seen this before, is this the exact formula written when stating the formula used in general terms is ...? – user1394925 Mar 30 '13 at 23:21
@user1394925: Not quite exact, since there are more decimal places, but yes, apart from rounding that’s the exact percentage. – Brian M. Scott Mar 30 '13 at 23:21
@user1394925: If $A$ and $B$ are independent events, the probability that both occur is the product of their individual probabilities of occurring. In this case you have $50$ events rather than $2$, but the principle is the same. – Brian M. Scott Mar 30 '13 at 23:22
@user1394925, the exact decimal is $0.000000000000088817841970012523233890533447265625\%$, if you really need it. – George V. Williams Mar 30 '13 at 23:25

Note that the probability of randomly choosing and answer/guessing and getting any ONE answer correct is $50\% = 1/2$.

But to get all $50$ answers correct, if guessing on each, the probability of getting $100\%$ is

$$\underbrace{\dfrac 12 \times \dfrac 12 \times \cdots \times \dfrac 12}_{\large 50\;\;\text{times}} \quad = \quad \left(\frac 12\right)^{50}$$

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Yes that is correct in what you are saying, as comment to previous user what is this formula called exactly? – user1394925 Mar 30 '13 at 23:22
This is simply the product rule: the probability of answering any one question each is independent of the the probability of answering any other question correctly. – amWhy Mar 30 '13 at 23:24

The probablility would be 1/2 for each question, you would multiply (1/2) by itself 50 times. That would equal 8.882*10^-16, a very small number. However, this would not be a real-life situation, because most of the time you would have at least some knowlege of the quesitons.Also, you would use guessing to eliminate incorrect choices, and that might increase your chances of getting a correct answer above the number mentioned.

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