Do you know about binomial coefficients?

, the number of combinations of

*N* objects taken

*k* at a time?

It is also the number of ways the arrange

*N* 1's and

*k* 0's in a string.

We think of the 1's as a place where a success occurs and the 0's where a failure happens.

If success has a probability of

*p* then failure has probability of

*1-p*.

So the

__probability of exactly__ *k* successes in

*N* trials is

.

Thus on a test of ten questions with four options, a student just guesses.

The

__probability of exactly__ *4* successes in

*10* is

.

For at least four is

.