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October 14th, 2019, 06:56 AM   #1
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Probability of correct answers with multiple choices

Hello;
This is probably an elementary question for most people, but I have a question. Let's say that there is a 100 question test with each question having two possible choices. With an ideal random number generator the probability getting the correct answer approaches 50% as the number of questions increases. That I know. My question is this. For a person trying to get the correct answer, is it possible to score less than 50%? How much less?
Thank You Very Much;
Frank
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October 14th, 2019, 07:27 AM   #2
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It's certainly possible for them to get zero correct. It's not likely but it's possible.

You might read up on the binomial probability distribution.
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October 14th, 2019, 07:30 AM   #3
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Yes, it is possible to score less than 50 %. It is possible to score 0 % (although there is a 1 in 2^100 chance of doing so). In fact, you have the same odds of scoring less than 50 % as you do of scoring more than 50 %.

What is more interesting to look at is: What is the probability of scoring 50 %? or 90 %? or 25 %?

In this case, we have a binomial distribution
$\displaystyle p(n) = \left[ \frac{N!}{(N-n)!n!} \right] R^n (1-R)^{N-n},$
where N is the number of questions (100), n is the number of questions you guessed correctly (i.e., the score), and R is the probability of guessing any given question correctly (i.e., 0.5).

In this case, the probability of making a score of n/100 is
$\displaystyle p(n) = \frac{100!}{(100-n)!n! 2^{100}}$

We can also calculate the cumulative distribution function P(n). This is the probability of making a score of n or lower.
$\displaystyle P(n) = \sum_{i=0}^{n} p(n)$

With N = 100 questions, there is a 7.96 % chance of scoring exactly 50/100. There is a 46.02 % chance of scoring lower than this, and a 46.02 % chance of scoring higher than this.
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October 14th, 2019, 07:32 AM   #4
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We need a, "Someone posted a response while you were typing," warning on the forum, lol.
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