Compound Events - Probability, Sample Space & Independent Events

# Compound Events

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 probability of a simple event and a compound event can be calculated in the same way, that is, the ratio of number of favorable events to the total number of events in the sample space. The sample space of a compound event is little complicated to calculate. For example, rolling a die is a simple event but rolling a die and tossing a coin simultaneously is a compound event.

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

To find out sample space of compound events is a little complicated process. For example, if we have an event of picking a card from deck of 52 cards and tossing a coin then how large will be the sample space will be. And it can create a lot of confusion also. There are three ways to create a sample space for a compound event.

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

Independent events are ones where the probability of occurrence of one event is not affected by the other one. The compound events consists of independent events also. For example, tossing of coin and rolling a die are independent events. Now, the probability of getting a 2 and a head can be calculated both ways. Either by any of the three methods we can find that total number of events in sample space is 12. The probability of getting 2-H is $\frac{1}{12}$.

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 $\frac{1}{2}$ $\times$ $\frac{1}{6}$ = $\frac{1}{12}$.