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

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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 #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 #1/6#
Roll a "2": Probability is #1/6#
Roll a "3": Probability is #1/6#
Roll a "4": Probability is #1/6#
Roll a "5": Probability is #1/6#
Roll a "6": Probability is #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:
#1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1#,
so the second property of a probability distribution holds true.

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