Representations of Probability

I'm working on the assumption that most folks at this point in their math career know how to calculate simple probability like the chance of rolling an even number on a 6-sided die (i.e. 1/2). If you can't, well, sorry.

### Notation

Notation for probability in the IB can be a bit inconsistent you need to be able to use and more importantly read probability problems when the info is given in different ways. Often problems are stated clearly such as "the chance that Nancy will buy a blue car is 1/4." Other times the IB (rightfully) use more "mathy" notation.

To demonstrate imagine you are rolling a die and the outcome your are interested in is the rolling of an 4 and for the sake of obscurity we call that event A. The probability of event A occurring can then be written as:

(1)
\begin{align} P(A)=\frac{1}{6} \end{align}

Pretty straight forward. This same probability can be written as:

(2)
\begin{align} P(X=4) = \frac{1}{6} \end{align}

This notation goes one step further to indicate the outcome we are talking about is the outcome of "4."

### Probability Distribution Tables

These are pretty straight forward once you see one. The table is just a way of writing down the probability of all potential outcomes of a particular event.

For example lets image we flip a coin twice. The possible outcomes are HH, HT, TH, TT. In this case we'll treat HT and TH as the same outcome of one head and one tail. So the probability distribution table looks like:

 Outcome 2 Heads 1 Head and 1 Tail 2 Tails Probability $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{4}$

Note that the probabilities must always add to one! A common IB problem is to give the student an incomplete table to fill in. Often to find the last bit of info you must use the tidbit that all the probability add to one.

### Venn Diagrams

Venn Diagrams are simple and awesome. I often find them the key to working through a tough problem. Even if the problem doesn't ask you to create one they can often be helpful. Venn diagrams are essentially a tool to help visualize and categorize potential outcomes.

For example imagine we were pulling a random card out of a deck and were interested (I use that word loosely) if that card is either Black or Even. We have created two circles to indicate the outcomes of Black and Even. Since there are some black cards that are also even the circles overlap.

This diagram indicates that there are 26 total (18 + 8) black cards of which 8 are also even. It also shows that there is a total of 16 even cards of which 8 are black and 8 are not black (i.e. red). The diagram also shows that there are 18 cards that are neither black nor even.

From this info all kinds of questions can be asked and answered with relative ease.

Venn Diagrams are great when two outcomes can occur at the same time, such as a card being black and even.

### Tree Diagrams

Tree diagrams can be tedious and hard to organize, but can also be very useful. They are best used when examining events that are mutually exclusive (i.e. heads or tails) and when there is a maximum of three potential outcomes and only 2 or 3 repetitions. These conditions are fairly common on IB questions.

Imagine that we roll a die twice and we define Event A to be rolling a 1,2,3 or 4 and Event B to be rolling a 5 or 6. The diagram would then look like: On the diagram the fractions represent the probability going down that branch. In the case above the events are independent so the probabilities of events are not affected by the prior event.

From the tree diagram we can calculate the probability of getting A twice as:

(3)
\begin{align} P(A)P(A)=\frac{2}{3} \frac{2}{3} = \frac{4}{9} \end{align}

Even more useful we might be curious of the probability of getting event A and B but in any order. That corresponds to getting A and B or B and A which are both branches on the tree diagram.

(4)
\begin{align} P(A \cup B) = P(A)P(B) + P(B)P(A) = \frac{4}{9} \end{align}

### Table of Results

Another useful way to organize data is what I'm calling a table of results. For example say we want to investigate the probabilities involved in the sum of two die - say a red die and a green die. We can do that in a table: When there are two events and you are interested in the combination of those outcomes this kind of table can be very useful. If we wanted to know the chance that the sum was 8 we could simply count the number of 8's and the total outcomes to get an answer of:

(5)
\begin{align} P(X=8)=\frac{5}{36} \end{align}