Discrete probability distribution

< List of probability distributions

A discrete probability distribution is a set of probabilities associated with the outcomes of a random variable. Random variables can be either discrete or continuous, meaning that they may take on values that can only be counted (discrete) or values that can be measured (continuous). Discrete variables are those which have a finite number of possible values. Examples include the outcome of a coin toss (heads or tails), the result of a dice roll (1 through 6), or the number of people who visit a website in one day (any integer from 0 to infinity).

discrete probability distribution
A discrete probability distribution graph, showing a series of individual data points.

Discrete probability distribution example

Discrete probability distributions are associated with discrete variables where each value has an associated exact probability. For example, if you were to flip a fair coin once, the probability that it will land on heads is 0.5 and the probability that it will land on tails is also 0.5. In other words, if you were to flip the coin an infinite number of times, approximately half would come up heads and half would come up tails. This relationship between each discrete value and its corresponding probability is called a discrete probability distribution.

Let’s say you had the choice of playing two games of chance:

  • Rolling a die (if you roll a 6, you win a prize) and
  • Guess the man’s weight: if your guess is correct to within 10 pounds, you win a prize.

Which of these games is a discrete probability distribution and which is a continuous probability distribution?

For the rolling the die game, each number on a die has an equal chance of being rolled. This gives you a discrete probability distribution, which you can summarize in a table:

Roll a…123456

For the guess the weight game, you could guess 150 pounds, 121.5 pounds, 67.878 pounds… the possibilities are endless:

  • 150.1 lbs.
  • 150.11 lbs.
  • 150.111 lbs.
  • 150.1111 lbs.
  • 150.111111 lbs.

The fact that you can theoretically guess an endless (infinite) number of weights, means that this is a continuous probability distribution.

Discrete Probability Distributions in Practice

Discrete probability distributions are helpful when making certain predictions related to random events. For example, if you wanted to estimate how many visitors your website would have in one month based on historical data, you could use a discrete probability distribution to calculate an expected value—the average number of visitors per day over time—and then multiply that by 30 days to get your estimated monthly total. This type of prediction is useful for businesses who want to plan their budget accordingly or make decisions about hiring more staff based on expected customer demand.

Common discrete probability distributions include:

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