# Binomial Probability, Real Limits, X=0

#### jimbrowsky

##### New Member
Can someone have guessed on quiz and got all 50 questions wrong, but still get 100%?

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#### Dason

Does it seem plausible that a student that just guessed would get all 50 wrong?

#### Dason

Well you probably need math to show that it's VERY improbable of getting them all wrong just by guessing.

Would you give somebody who got them all correct an A? Why would you do that? Because they knew the material well enough to answer all of the questions correctly. If they get them all wrong then it seems like they did in fact know the correct answer but they answered them all wrong on purpose. So they still knew the material well enough that they could have answered them all correct. Or maybe they thought they were answering them correctly but filled in the wrong circle on a scantron or something... It's silly but that does seem to be the implication.

#### BGM

##### TS Contributor
BTW one point to add: to calculate the probability mass function $$\Pr\{X = 0\}$$ there is no need to use normal approximation. It is just as simply as $$\frac {1} {2^{50}}$$.

#### Dason

1/2^50 = $$\frac{1}{2^{50}} \approx \mbox{8.88e-16} = 8.88*10^{-16} = .00000000000000088$$

I'm thinking you didn't notice the e-16 on your calculator (or you didn't understand it?)

1/2^50 is just the exact binomial probability for X=0 when you plug the right numbers into the binomial pmf.

#### Dason

I think you're confused on what X is? X is the observed number of successes. So inherently it must be an integer. I think what you're trying to describe is P(X = 0) which is related to but is different than X.

#### BGM

##### TS Contributor
I dont understand your last point. Maybe I'm not that far into stats to comprehend 1/2^50? You mean, 1/2^50 as the probability for getting each T/F question correct (or false)?
I suppose you have learned the concepts about independence before you learned the binomial distribution (and of course should before you learned the normal approximations)

So if you used the independent argument, even though you know nothing about Binomial distribution, you can still come up with that answer.

Of course the statement "What is the probability of getting a consecutive 50 failures in 50 independent and identical trials ?" can be in general viewed as a Binomial distribution problem.

#### BGM

##### TS Contributor
"What is the probability of getting a consecutive 50 failures in 50 independent and identical trials?" We need to provide the z-score
So that is exactly the thing I want to talk about. To evaluate such probabilities does not require normal approximation or even knowing Binomial distribution. All you need to know is the independent properties. I suggest you first to know what is random variable, Binomial distribution first. Then try to understand why people need to make use of normal approximation, and when you need such approximation.