"Bernoulli's Backdoor"

~Here's something that will be very helpful when solving binomial distribution problems involving Bernoulli trials. I called it "Bernoulli's Backdoor", since it involves the "backdoor approach" to probability. Here's how it works.

~We know, for a Bernoulli trial, we can calculate the probability of a number of successes. We need to know n (# of trials), probability of a success (p), and the number of successes we want (r). Just use 2nd, VARS, find it on the menu, insert these 3, then Enter.

~However, for problems with "at least"or "at most" a given number of successes, we need to do it more than once,then add. (i.e., "at least 3" means 3 or more). ("at most 3" means 3 or less).

~If the number of trials is large compared to the number of successes we want, this could force us to repeat the formula many times (i.e., at least 3 successes in 30 trials).

~That could be extremely time consuming & also increase the likelihood of computational errors.

~So, use Bernoulli's backdoor. That is, find the probability of the negative (complement) of what we want, then subtract it from one. (i.e., for at least 3 successes in 30 trials, it would be
1 - p(less than 3 successes in 30 trials).

~That way, you only have to use the formula once (using 2nd VARS, A key on the menu...gives the cumulative probability up to 2 successes), however, you must remember to subtract the result from 1. You can compute the the final result with one calculation by doing the following:

Enter 1 - 2nd Vars, menu A, (number of trials, probability of success, upper limit of the cumulative number of trials). For this problem, enter
1 - 2nd Vars, menu A, enter (30, p, 2).

~Some people might not have to use the formula at all, since the combinations are small & can be found easily & the other parts can be calculated simply by the power key.

~Keep this in mind, if you see one like this on a quiz or exam.

~For the "at most" problems, the backdoor approach is not used since it is already in the cumulative state (i.e., starts at 0 up to the given number of successes). So, menu A is used directly without subtracting from one.  For example, to get "at most 7 heads" when flipping a fair coin 20 times, just go to menu A & insert (20, .5, 7).