Suppose you are taking a test of 100 questions, each of which has 4 options and values 1 point, and there is only 1 option is the correct answer. The pass line is set to 60 points. Now, you are holding a lottery of A, B, C, D to guess the correct answers all by chance.
Is it possible to pass the test all by chance?
Let’s think of it by serious mathematics.
First, because you try to choose a correct answer for each question, the possibility you succeed is . And there is no doubt that the choosing action is a Bernoulli Experiment. So it is adapted to Binomial Distribution. Denote the random variable for a successful choice with .
The probability we are chasing is:
In general, successes in choices,
And, the cumulative distribution is:
So in our problem here,
Obviously, it is pretty hard to calculate this in that form.
But when is big enough, we can transform the Binomial Distribution to Normal Distribution, by the famous “de Moivre–Laplace theorem“. So approximately,
By the way, a Normal Distribution density function is:
And approximately, what we should do in our problem is, get the result of:
In order to look up in a Normal Distribution table, transform it to the Standard Normal Distribution format.
Finally, use Standard Normal Distribution Table to find the result of this:
.
Ok, you see it, it is wise to study hard and do not play a lottery in your test. It does not help anyway.