Bernoulli trials

01Model one binary outcome#

A Bernoulli variable records one event as $1$ when it occurs and $0$ when it does not. Its parameter $p$ is the probability of the event, with $0\le p\le1$.

For a binary classifier's correctness indicator, $X=1$ can mean correct and $X=0$ incorrect. If its correctness probability under a stated sampling model is $0.8$, then $P(X=1)=0.8$ and $P(X=0)=0.2$.

02State what one means#

Writing $X\sim\operatorname{Bernoulli}(p)$ specifies the two possible values and their probabilities. The meaning of success is chosen by the model; it need not be a desirable outcome. You could instead define $1$ as an error, changing the parameter.

If $1$ means error with probability $0.2$, the two masses are $0.2$ at one and $0.8$ at zero. The underlying events have not changed, but the numerical encoding has.

03One outcome versus a count#

A count of successes across several trials generally takes values beyond zero and one, so it is not a single Bernoulli variable. Its individual trial indicators may be Bernoulli variables. Independence assumptions belong to a model of the collection, not to the definition of one indicator.

At $p=0$, the variable is always zero. At $p=1$, it is always one. Both boundary cases are valid distributions. A probability outside the interval from zero to one is invalid.

When using a Bernoulli model for a binary label, say what the label represents and which population or condition defines $p$. One binary observation does not establish the probability by itself.