Union and Intersection

Compound events are formed from several sample points belonging to different events. These operations can be described by set theory and its operators. These operations can be nicely visualized and easily understood by using the so-called Venn diagram. The most basic operators are the union and intersection.

Union

The union of the two sets A and B is defined as the set of all events which belong either to set A or to set B. The union corresponds to the logical OR and is denoted using the symbol

Intersection

The intersection of the two sets A and B is defined as the set of events which belong to both A and B. The intersection corresponds to the logical AND and is denoted using the symbol .

Example: Let us take a quick look at the game Roulette (the European version with only one zero field):

How many numbers let us win, if we bet on the red numbers and the numbers of the third column? In order to evaluate this we have to calculate the union of the set A (the red numbers) and the set B (the numbers in the third column). A  B contains 22 numbers, which are comprised of 18 red numbers and the 12 numbers in the third column. It is evident that the nnumber of elements of the union A  B is smaller than the sum of the individual sets, since some of the numbers belong to both sets.

Set	Elements	Probability
A	{1,3,5,7,9,12,14,16,18,19, 21,23,25,27,30,32,34,36}	18/37 = 0.4865
B	{3,6,9,12,15,18,21,24,27,30,33,36}	12/37 = 0.3243
A  B	{1,3,5,6,7,9,12,14,15,16,18,19, 21,23,24,25,27,30,32,33,34,36}	22/37 = 0.5946
Note:	The calculation of the probability has to be based on 37 numbers because of the zero field.

How many numbers win twice if we bet on red and the third column? In order to win twice the selected number has to be an element of the intersection of set A (red numbers) and set B (numbers in the third column). The insection contains 8 numbers, thus we have a chance of 8/37 to win twice.

Set	Elements	Probability
A	{1,3,5,7,9,12,14,16,18,19, 21,23,25,27,30,32,34,36}	18/37 = 0.4865
B	{3,6,9,12,15,18,21,24,27,30,33,36}	12/37 = 0.3243
A B	{3,9,12,18,21,27,30,36}	8/37 = 0.2162

The probabilities of compound events can be calculated by counting the sample points and applying the summation rule. In addition, there exist some rules to calculate the probabilities for more complex problems which do not allow to count the sample points.