Events can be defined as subset of a sample space S. Events can be broadly classified as simple events and compound events. A simple will have a single outcome and will show occurrence of a single event. A compound event is the occurrence of two or more events together. The main complication to get the probability of compound events is to get the sample space as the sample space becomes uncomfortable large at times.
Compound Events in Probability
The sample space of rolling a die = {1, 2, 3, 4, 5, 6}
The sample space of rolling a die and tossing a coin together = {(1, H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T), (5, T), (6, T),}
Compound Events Sample Space
1) Ordered List: The events can be written as a list. For example, the sample space of a compound event of rolling a die and tossing a coin can be written as,
1-H, 2-H, 3-H, 4-H, 5-H, 6-H, 1-T, 2-T, 3-T, 4-T, 5-T, 6-T
2) Table: A table is a way to representing the sample space of a compound event and it does so very clearly. The sample space of rolling a dice and flipping a coin can be represented in a table as given.
H | T | |
1) |
1-H | 1-T |
2) | 2-H | 2-T |
3) | 3-H | 3-T |
4) | 4-H | 4-T |
5) | 5-H | 5-T |
6) | 6-H | 6-T |
3) Tree Diagram: A tree diagram shows the events of a sample space as the leaves of the tree diagram.
Compound Events & Independent Events
If we take two independent events then probability of getting a head is $\frac{1}{2}$ and the probability of getting a 2 is $\frac{1}{6}$. Hence, probability of getting a head and a 2 is $\frac{1}{2}$ $\times$ $\frac{1}{6}$ = $\frac{1}{12}$.