# Properties of a Binomial Experiment

## Key Questions

• In a Binomial setting, there are only two possible outcomes per try. Depending on what you want, you call one of the possibilities Fail and the other one Succes.

Example :
You may call rolling a 6 with a die Succes, and a non-6 a Fail. Depending on the conditions of the game, rolling a 6 may cost you money, and you may want to reverse the terms.

In short:
There are only two possible outcomes per try, and you may name them as you want: White-Black, Heads-Tails, whatever.
Usually the one you use as $P$ in calculations is called (probability of) Succes.

• In a BInomial setting there are two possible outcomes per event.

The important conditions for using a binomial setting in the first place are:

• There are only two possibilities, which we will call Good or Fail
• The probability of the ratio between Good and Fail doesn't change during the tries
• In other words: the outcome of one try does not
influence the next

Example :
You roll dice (one at a time) and you want to know what the chances are that you roll at lest 1 six in 3 tries.
This is a typical example of binomial:

• There are only two possibilities:
6 (chance $= \frac{1}{6}$) or not-6 (chance $= \frac{5}{6}$)
• The die has no memory, so:
• Every next roll has still the same probabilities.

You can set up a chance-tree, but you can also calculate the chance of three Fails, which is

$\frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} = \frac{125}{216}$

And your chance of succeeding would be

$1 - \frac{125}{216} = \frac{91}{216}$