Probability distributions are tables that give the probabilities for things like rolling a die, or tossing a coin.  One can construct one using the tree diagram for tossing the coin previously shown.

Say we want to construct a probability distribution for a variable H representing the number of heads that occur in three tosses.

                                H = number of heads that occur in three tosses

We can make a table as follows:

Note how the probability column adds up to 1.  Probability distributions should add up to 1. Sometimes this information is shown in a graph instead of a table.

Binomial distributions

Binomial distributions come about from situations where there are only two possible outcomes – such as tossing a coin, when you can only get heads or tails.

So far we can do questions like:

If I toss a coin three times, what is the probability of getting 2 heads then a tail?

To answer this question, we could just look at our tree diagram and see that it is 1/8.

What about:

If I toss a coin three times, what is the probability of getting exactly 2 heads?

This is more complicated, because we could get the following combinations which would all be counted as two heads and a tail:

                                                               HTH

                                                               HHT

                                                               THH

The overall probability of this occurring would be .

Another way of doing this is to use a general expression for binomial probability (‘bi’ meaning ‘two’):

                                  Probability of x desired successes in n trials

                          = number of possible ways this can occur .

‘s’ is the probability of a (s)uccess happening in any one trial

‘f’ is the probability of a success not happening in any one trial, it is the probability of (f)ailure in any one trial.

To illustrate this, let’s use a coin example again.

Binomial distribution question

What is the probability of two heads exactly occurring in three tosses?

Solution

A success is a head occurring. The desired outcome is two successes exactly. So:                                                                x = 2 The number of trials is 3. Number of possible ways this can occur is 3 (HTH, THH, HHT).                This confirms our previous answer.