Venn diagrams
In order to use Venn diagrams when talking about events, we must first understand the term 'mutually exclusive'. Imagine there are two events: event A and event B. If they both cannot happen at the same time then A and B are mutually exclusive.
Example
Jim is going to roll a dice.
Let’s say that event A is rolling an odd number and event B is rolling the number 2. Quite clearly these events are mutually exclusive because you cannot roll both a 2 and an odd number with a single roll of a dice.
This is represented on a Venn diagram like this:
The fact that the two circles do not overlap shows that the two events are mutually exclusive.
This means that the probability of A or B happening = the probability of A + the probability of B.
This is written as P(A or B) = P(A) + P(B).
Consider a second example where event A is chosen to be getting an even number and event B is chosen to be getting a number greater than 3. These events are not mutually exclusive because the criteria for both is fulfilled if we roll a 4 or a 6.
The 4 and 6 are placed in the overlapping middle quadrant as they represent outcomes which satisfy both events.
This means that the probability of A or B happening = the probability of A + the probability of B – the probability of A and B.
P(A or B) = P(A) + P(B) – P(A and B).
Let’s see if this is correct:
P(A or B) means the probability of getting an even number or a number greater than 3. This means we succeed if we get {2,4,6} or {4,5,6}, so in other words we succeed if we get {2,4,5,6}.
The probability of this is \(\frac{4}{6}\).
P(A) means the probability of getting an even number. This means we succeed if we get a {2,4,6}, so the probability is \(\frac{3}{6}\).
P(B) means the probability of getting a number greater than three. This means we succeed if we get {4,5,6}, so the probability of this is also \(\frac{3}{6}\).
P(A and B) means the probability of getting an even number that is also greater than 3. This means we succeed if we get {4,6}, so the probability is \(\frac{2}{6}\).
\(\frac{4}{6}~=~\frac{3}{6}~+~\frac{3}{6}~-~\frac{2}{6}\). So the formula works.
In the previous section, we used the notation P(A and B) which is called A intersection B. The outcomes which satisfy both event A and event B, this is written P(A ∩ B) and is the overlapping area on the Venn diagram.
We also used the notation P(A or B) which is called A union B, the outcomes which satisfy either event A or event B, this is written as:
P(A ∪ B)
It's represented by the two circles including the overlapping section on the Venn diagram.
It is important to note that all the outcomes that do not satisfy event a are written as A’. This is said to be the complement of A, and is represented by all the area outside of circle A on the Venn diagram.
Question
A bag contains 4 green balls, 3 red balls and 7 balls of other colours. Draw a Venn diagram for this information.
Question
Draw a Venn diagram for the outcome of a rugby match where event A is that the home side scores a try and event B is that the home side wins the match.
Notice that these events are not mutually exclusive because you can score a try and win a match. In fact, it’s quite common. However, it is also possible to score a try and still lose. It is also possible to win without scoring any tries.