# Probability Distribution: List of Statistical Distributions

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## What is a Probability Distribution?

A probability distribution tells you what the probability of an event happening is. Probability distributions can show simple events, like tossing a coin or picking a card. They can also show much more complex events, like the probability of a certain drug successfully treating cancer.

There are many different types of probability distributions in statistics including:

The sum of all the probabilities in a probability distribution is always 100% (or 1 as a decimal).

### Ways of Displaying Probability Distributions

Probability distributions can be shown in tables and graphs or they can also be described by a formula. For example, the binomial formula is used to calculate binomial probabilities.

The following table shows the probability distribution of a tomato packing plant receiving rotten tomatoes. Note that if you add all of the probabilities in the second row, they add up to 1 (.95 + .02 +.02 + 0.01 = 1).

The following graph shows a standard normal distribution, which is probably the most widely used probability distribution. The standard normal distribution is also known as the “bell curve.” Lots of natural phenomenon fit the bell curve, including heights, weights and IQ scores. The normal curve is a continuous probability distribution, so instead of adding up individual probabilities under the curve we say that the total area under the curve is 1.

Note: Finding the area under a curve requires a little integral calculus, which you won’t get into in elementary statistics. Therefore, you’ll have to take a leap of faith and just accept that the area under the curve is 1!

## Probability Distribution Table

A probability distribution table links every outcome of a statistical experiment with the probability of the event occurring. The outcome of an experiment is listed as a random variable, usually written as a capital letter (for example, X or Y). For example, if you were to toss a coin three times, the possible outcomes are:

TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

You have a 1 out of 8 chance of getting no heads at all if you throw TTT. The probability is 1/8 or 0.125, a 3/8 or 0.375 chance of throwing one head with TTH, THT, and HTT, a 3/8 or 0.375 chance of throwing two heads with either THH, HTH, or HHT, and a 1/8 or .125 chance of getting three heads.

The following table lists the random variable (the number of heads) along with the probability of you getting either 0, 1, 2, or 3 heads.

 Number of heads (X) Probability P(X) 0 0.125 1 0.375 2 0.375 3 0.125

Probabilities are written as numbers between 0 and 1; 0 means there is no chance at all, while 1 means that the event is certain. The sum of all probabilities for an experiment is always 1, because if you conduct and experiment, something is bound to happen! For the coin toss example, 0.125 + 0.375 + 0.375 + 0.125 = 1.

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## More complex probability distribution tables

Of course, not all probability tables are quite as simple as this one. For example, the binomial distribution table lists common probabilities for values of n (the number of trials in an experiment).

The more times an experiment is run, the more possible outcomes there are. the above table shows probabilities for n = 8, and as you can see — the table is quite large. However, what this means is that you, as an experimenter, don’t have to go through the trouble of writing out all of the possible outcomes (like the coin toss outcomes of TTT, TTH, THT, HTT, THH, HTH, HHT, HHH) for each experiment you run. Instead, you can refer to a probability distribution table that fits your experiment.

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## List of Statistical Distributions

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