Understanding Probability Distribution

The probability distribution of a particular random variable indicates the relative frequency of each of the potential outcomes for that variable. Fortunately, the concept of probability distribution is based on simpler topics. After gaining an understanding of probability, random variables, and relative frequency, probability distribution becomes easier to understand.


Probability simply refers to the likelihood of a particular event occurring. The event in question can be anything from the chance of winning in blackjack to the likelihood of a coin landing on its head. Probabilities can be expressed in percentages or in fractions. For example, there is a 50% chance that a coin will land on its head and a 1/4 chance that someone will pick a hearts card from a new deck. The desired outcome can also combine multiple criteria, such as the probability of rolling an even number over three on a die.

Random Variable

A random variable is any outcome which can have more than one result. Examples of random variables include throwing a die, picking a card, or flipping a coin. Note that the action itself is not the random variable; the result is. In the case of a coin which is flipped two times, the random variable would be whether the coin landed on its tail or head. X would represent all possible outcomes (heads tails, heads heads, tails tails, tails heads), whereas x would represent the actual outcome of the trial, such as heads tails.


The frequency of a variable refers to how many times the outcome occurs. For example, if a coin was flipped ten times and six of those times it landed on its head, the random variable of heads would have a frequency of six. Frequencies in and of themselves are not always helpful since they do not explain the big picture. Picking the ace of hearts five times on five trials is much more unlikely than picking it five times in a thousand trials.

Relative Frequency

The relative frequency of an outcome refers to the number of times the outcome occurred relative to the total number of trials. In the example above, the relative frequency of heads would be 6/10 or 60%. Relative frequencies are always given as a percentage, a decimal point, or a fraction. The sum of the relative frequencies of all the outcomes is always 100%, or 1. A relative frequency of 0.6 for heads would mean a relative frequency of 0.4 for tails, which would total one.

Putting It All Together; Understanding Probability Distribution

The probability distribution usually takes the form of a graph, chart, or some other kind of visual aid. It can show the results of past trials or predict the likely results of future experiments, and it can indicate the actual frequency or the relative frequency. If a coin is flipped, the probability distribution will have one column for heads and one for tails, and the chart would show how often the coin landed on each one. The probability distribution is an easy and convenient way to visualize how often each result happened relative to the other results and the total number of trials.

At the beginning of this article, probability distribution was defined as demonstrating the relative frequency of each potential outcome for a random variable. Probability, random variable, and frequency are all fairly easy concepts on their own. Once those are explained and understood, the definition of probability distribution becomes much more manageable.

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