In probability formula, what does n represent?

[stepheckert]

In the formula,

PN(n) = N!/(n!(N-N)!) (p^n)(q^(N-n))

N represents the number of elements, but what exactly does little n represent?

Thanks for any help!

 

[CompuChip]

The formula you gave,
$$P_N(n) = \frac{N!}{n! (N - n)!} p^n q^{N - n}$$
is the probability of n successes after N trials in a binomial distribution.

That is, you run N trials and for each of them the chance for "success" is p, and the chance for "failure" is q = 1 - p. The formula above gives the probability that out of the N trials, you get $0 \le n \le N$ "successes".

Some examples: if you flip a coin N times, and define "success" as "heads comes up" (with p = 1/2) then P_N(n) is the probability of finding exactly n heads and N - n tails in N flips.

If you throw a die N times, and define "success" as "six comes up" (with p = 1/6) then P_N(n) is the probability of getting exactly n sixes and N - n other values in N throws.

If a multiple-choice exam has N questions, and you randomly select one of the four answers for each of them then "success" can be "question is answered correctly" (with p = 1/4) and P_N(n) is the probability of getting exactly n questions right and N - n wrong.

 

[stepheckert]

This helps so much! Thank you!