The field of probability is all about transforming counts of possible outcomes into geometric areas, volumes, etc.

Let \(S\) be a set called the *sample space*. Each element of \(S\) is called an event, \(E\). The events each have a value which ranges from \(0\) to \(1\) and all the elements of \(S\) must sum to \(1\).

Suppose we're rolling a fair die. There are six, discrete outcomes – I could roll a number anywhere from \(1\) to \(6\) and each outcome is equally likely. Suppose we rolled the die some arbitrary number of times and we saw that the distribution was *not* \(\frac{1}{6}\)th for each outcome. We would conclude that the die was *weighted*, or otherwise known as not fair.

Let's see what happens when we're working with an unfair set of dice. We can write a function for a fair die.

```
function rollfairdie()
rand(1:6)
end
function sumfrequencies(samplespace, events)
frequencies = zeros(Float64, length(samplespace))
for evt in events
frequencies[evt] += 1
end
frequencies
end
```

Here we're generating a random integer between one and six, equally.

```
rolls = 100
events = [ rollfairdie() for _ in 1:rolls ];
probabilities = map(x -> x/rolls, sumfrequencies(collect(1:6), events))
bar(1:6, probabilities, legend=:topleft, label="Probabilitiy")
```

Given one-hundred rolls, we can see that the distribution is fairly evenly spread across each of the numbers one through six.

Now we'll use a feature from the `StatsBase`

package – `sample`

and `Weight`

. `sample`

takes two vectors, one is going to be the sample space, e.g. \(1\) to \(6\). The second one is the `Weight`

object which contains a associated value for each element in the first vector – the weight.

```
function rollweighteddie(; wts=Weights([1, 1, 1, 1, 6, 6]))
sample(1:6, wts)
end
rolls = 100
wts = Weights([1, 1, 1, 1, 6, 6])
events = [ rollweighteddie(wts=wts) for _ in 1:rolls ];
probabilities = map(x -> x/rolls, sumfrequencies(collect(1:6), events))
bar(1:6, probabilities, legend=:topleft, label="Probability")
```

Now we can see that since we've weighted the die with an higher likelihood of rolling a 5 or 6, and the plot shows us just that.

Suppose that we have two events such as I roll a die twice.

© Daniel Marvin. Last modified: July 19, 2021.