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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. 
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 $\displaystyle p(n) = \left[ \frac{N!}{(Nn)!n!} \right] R^n (1R)^{Nn},$ 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 $\displaystyle p(n) = \frac{100!}{(100n)!n! 2^{100}}$ 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. 
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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answers, choices, correct, multiple, probability 
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