Venn diagrams

(review probability notation here)

Venn diagrams are introduced here to help visualize some bases rules of probability. In a Venn diagram the squared area denoted e contains all possible events. The circles represent specific events. Probabilities are represented by the ratios of areas. For example, the probability of event A in the figure below is the ratio of area A to the total area e:

Figure 1: Venn diagram for a single event A

Mutually exclusive events

Figure 2 gives an example of a Venn diagram where two events (A and B) are identified. The events are mutually exclusive meaning that they cannot occur together, and therefore the circles do not overlap. The areas of the circles are denoted A and B, and the probability of the occurrence of events A and B. are denoted P(A) and P(B) given by:

P(A) = A/e

P(B) = B/e

Figure 2: Venn diagram for two mutually exclusive events

You can think of a Venn diagram as an archery target. Imagine that you are firing an arrow at the target and that you have equal chance of landing anywhere within the target area, but will definitely hit it somewhere. The circles on the target represent each possible event so if your arrow lands in circle A, it represents event A happening. In Figure 2, you cannot fire an arrow that will land in both A and B at the same time, so these events cannot occur at the same time:

The probability of either event occurring is then just the sum of the probabilities of each event because we just need to add the A and B areas together:

Events that are not mutually exclusive

In Figure 3, A and B are not mutually exclusive: they can occur together, represented by the overlap in the Venn diagram. The figure shows the four different areas that are now produced. It can be seen from these areas that:

Figure 3: Venn diagram for two events that are not mutually exclusive

We know from basic probability rules that:

if A and B are independent, or

if they are not independent.

How is that represented in the Venn diagram?

A and B are independent

 is the area of overlap between the two circles. The probability of  is

Written in terms of areas, this would be:

In other words, the ratio between the overlap area and area A must be equal to P(B). Similarly, the ratio between the overlap area and area B must be equal to P(A).

A and B are dependent

Written in terms of areas, this would be:

In other words, the ratio between the overlap area and area A must again be equal to P(B|A).

More complex example

A Venn diagram is a great way of getting across the complex interactions between probability events, providing you are released from making the areas actually correspond to the required probabilities (because that gets very tedious). The following example is difficult to describe mathematically, but simple in a Venn diagram:

Figure 4: More complex Venn diagram example

Read on: Event trees

 

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