Binomial experiments

One tough part of probability is recognizing which rule to use and when. Binomial probabilities may seem difficult, but in a way they are nice because there is a set formula to use. However, to know to use this formula, you must first determine whether or not the situation you are working with represents a binomial experiment. We will see how to do this below.

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How to determine if an experiment is binomial

A binomial experiment is any probability experiment where the following four properties hold.

  • The experiment consists of a series of n trials.

    You can think of each trial as a “mini-experiment”. For example, if you flip a coin 10 times, there are 10 mini-experiments. The mini-experiment, or trial, is “flip a coin”. The number of trials is usually labelled “n”.

  • Each trial has two outcomes.

    We call these two outcomes “success” and “failure”, but really a success is simply the label for whatever we are counting. In the coin flip example, we could call heads a success and tails a failure.

  • There is a fixed probability of success for each trial.

    For an experiment to be a binomial experiment, we must have that the probability of success doesn’t change from trial to trial. Using a coin flip again (flipping a coin multiple times is a classic binomial experiment example), the probability of heads stays the same on each flip. The probability of success is usually labelled “p”, while the probability of failure is usually labelled “q”.

  • Each trial is independent of the others.

    It is said that a coin “has no memory”. That is, regardless of whether or not the coin landed on tails 100 times in a row, the probability of heads or tails remains the same. This idea of independence is needed in order to classify a probability experiment as being a binomial experiment.

Examples of binomial experiments

An important skill in a probability and statistics course is to be able to identify binomial experiments. So, we will look at a couple of examples and see how they fit the rules listed above.

Example

Data collected from a website shows that 39% of visitors use internet explorer. Randomly select 6 visitors and record how many use internet explorer.

  • fixed number of trials
    We can view this experiment as 6 repetitions of the mini-experiment “determine if a visitor uses internet explorer”. Thus, there are n = 6 trials.

  • two outcomes
    There are two possible outcomes to each trial: the visitor uses internet explorer or they do not use internet explorer.

  • fixed probability of success
    A success is often based on what you are recording. Here, we are interested in how many visitors use internet explorer. Given the data, we can assume that for any visitor, the probability they use internet explorer is 39%. So, p = 0.39.

  • independence
    There is no reason to assume that the probability one person uses internet explorer is affected by the fact that another person uses it. Thus, we do have independence.

Given the above information, we can say that this is a binomial experiment.

Example

Roll a single 6 sided fair die 14 times. Record the number of times a 5 comes up.

  • fixed number of trials
    We can view this experiment as 14 repetitions of the mini-experiment “roll a 6 sided fair die”. Thus, there are n = 14 trials.

  • two outcomes
    There are two possible outcomes based on what we are recording: “a 5 comes up”, “a 5 does not come up”.

  • fixed probability of success
    The probability a 5 comes up is 1/6. This is the same for each die roll. Thus, p = 1/6.

  • independence
    Die rolls, like coin flips, are always independent.

Once again, since these properties hold, we have a binomial experiment.

Why is this useful?

It may seem really academic to say that an experiment is binomial or not. However, it is actually very useful when it comes to calculating probabilities. Knowing an experiment is binomial allows us to use a formula or technology to find the probability of any number of successes occurring. To see this, take a look at this binomial probability example.

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