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Bayes' Theorem 1: Introduction
Hi, I'm Dr Nic, and in this video I am going to introduce Bayes theorem, and conditional probability.
Bayes theorem is an extremely important formula, which underpins a powerful concept.
Strictly speaking we should say Bayes's theorem as it is the theorem belonging to Thomas Bayes.
However both Bayes and Bayes's are used. I will use Bayes' as it is easier to say.
Here is Bayes Theorem, which is stated as a formula: The probability of A given
B = the probability of B given A times the probability of A all over the probability of B
We will illustrate this with some objects made out of coloured bricks.
Here are some objects. If we put them in a bag and draw one out without looking
what is the probability that it has a yellow brick?
We can see that there are nine objects with yellow, out of a total of 16 objects.
The probability of drawing one out that is yellow is nine out of 16.
Now say I can tell you that the object I drew out has a red brick.
What is the probability that it has yellow?
There are only five objects with red, so we only need to consider them.
two of the five with red have yellow and three have white.
So we say that the probability of Yellow GIVEN red is 2 out of 5 or two fifths.
What is the probability of red, given Yellow?
This means we know it is yellow, so we are only interested in the nine objects with Yellow.
Two are red and yellow, so our probability is 2 over nine or two ninths.
The other thing we need to consider is, what is the probability of Red And Yellow.
We are back to having no information about the object.
There are 16 objects, so each object has a one sixteenth probability of being picked.
Two of the objects are red and yellow, so the probability of Red AND Yellow is 2 out of 16.
This symbol is the intersection symbol and means "AND".
There are several ways we can represent this - using a table, a tree and a formula.
Here is a table summarising the make up of the objects.
We calculate: The probability of Yellow = 9/16
The probability of Yellow given red = 2/5
The probability of Red given Yellow = 2/9
The probability of Red AND Yellow = 2/16
Tables make calculating probabilities much simpler.
This tree shows the same data:
Probability trees are sometimes used for modelling
probabilities, but they are less useful in this type of exercise.
Now let us see how Bayes' formula can be used
Say we were not given all the information, but rather were told the following:
P(Yellow) = 9/16 and P(Red) = 5/16
And the probability of red given yellow = 2/9
Could we then work out the probability of Yellow given red? (The other way around.)
The answer is yes. We could construct a table like this one here.
I will show how to do this in a later video.
Or we could use Bayes formula.
We will replace A with Yellow and B with Red, giving us:
The probability of Y given R = the probability of Red given yellow times the probability of Yellow
all over the probability of Red
That equals 2/9 times 9/16, all over 5/16
That equals 2/16, divided by 5/16 Which equals 2/5.
This has been an introduction to Bayes' Theorem.
See in the description below for links to other videos about Bayes' Theorem, screening and probability.
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