# What is the sum of the probabilities in a probability distribution?

Jan 9, 2015

The sum of the probabilities in a probability distribution is always 1.

A probability distribution is a collection of probabilities that defines the likelihood of observing all of the various outcomes of an event or experiment. Based on this definition, a probability distribution has two important properties that are always true:

• Each probability in the distribution must be of a value between 0 and 1.
• The sum of all the probabilities in the distribution must be equal to 1.

An example: You could define a probability distribution for the observation for the number displayed by a single roll of a die. The probability that the die with show a "1" is $\frac{1}{6}$.

That's because there are six possible outcomes, and only one of those outcomes is a "1". Lets label the probabilities of all the possible outcomes for the single die.

Roll a "1": Probability is $\frac{1}{6}$
Roll a "2": Probability is $\frac{1}{6}$
Roll a "3": Probability is $\frac{1}{6}$
Roll a "4": Probability is $\frac{1}{6}$
Roll a "5": Probability is $\frac{1}{6}$
Roll a "6": Probability is $\frac{1}{6}$

Each probability is between 0 and 1, so the first property of a probability distribution holds true. And the sum of all the probabilities:
$\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1$,
so the second property of a probability distribution holds true.