Finding the probability of combined events requires creating diagrams and tables as it is slightly more challenging due to the fact that multiple probabilities need to be calculated.
Sample space diagrams
The table below is a way of displaying all the outcomes from throwing two dice and adding them together. This is called a sample space diagram.
When two dice are rolled and the space diagram is filled in, the following table will appear:
Using the table above, it is possible to calculate that the probability of getting the total of 2 is and the probability of getting a total 10 or more is .
Other combined events
When two four-sided dice are thrown and the numbers added together, the following space diagram can be constructed with all of the outcomes.
Calculating the number of outcomes
Using the table above, to calculate the probability of getting a score of 3 or 4 the following methods is required.
The first step if to write the probability of getting a score of 3 or 4 as P(3 or 4). This is also the same as the probability of getting a 3 added to the probability of getting a 4.
Mutually exclusive events
Mutually exclusive events are two events which cannot occur at the same time. For example, when throwing two six sided dice, if on die lands on 2, then it is impossible to get a total score of 10.
When calculating mutually exclusive events – probabilities of each event can be added together
When calculating an event which is not mutually exclusive – probabilities of the “overlap” should be subtracted
Independent events are so called because the outcome of one does not effect the outcome of the other
Combining probabilities using multiplication
The formula for a probability of two independent events happening at the same time is:
P(A and B) = P(A) x P(B), if there are more than two event it is expanded in the following way:
P(A and B and C) = P(A) x P(B) x P(C)
Points to note:
- In general for probability, “and” means to multiply
- In general for probability, “or” means add