Bernoulli Distribution

The Bernoulli distribution has an outcome space of two events {0, 1}, one of which happens with probability 1-p and the other which happens with probability p.

$$P(X=x) =\begin{cases}1-p,&x=1\\p, &x=0\\0, &x\not=0,1\end{cases}$$

A real-world system this distribution can model is a coin flip. Here, the two outcomes are heads (represented by 1) and tails (represented by 0). If the coin is fair, then p = 0.5.

Question

Which of the following can be reasonably modeled by the Bernoulli distribution?

(a) Rolling a 6-sided die

(b) Passing or failing your driver test

(c) The weather tomorrow

Answer

(b) is the correct answer because the outcome of a driving test is always either a pass or a fail. (a) and (c) both have outcome spaces that have more than two events in them; the die has six outcomes and the weather can be sunny, rainy, cloudy, snowy, etc. The Bernoulli distribution is not a good fit for these systems.