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 October 14th, 2019, 06:56 AM #1 Newbie   Joined: Oct 2019 From: Maryland Posts: 1 Thanks: 0 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
 October 14th, 2019, 07:27 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,638 Thanks: 1475 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. Thanks from idontknow and DarnItJimImAnEngineer
 October 14th, 2019, 07:30 AM #3 Senior Member   Joined: Jun 2019 From: USA Posts: 383 Thanks: 207 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. Thanks from idontknow
 October 14th, 2019, 07:32 AM #4 Senior Member   Joined: Jun 2019 From: USA Posts: 383 Thanks: 207 We need a, "Someone posted a response while you were typing," warning on the forum, lol.

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