12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the Universal Set all elements defined by that set. We often use Venn diagrams to display the relationships within sets and sample spaces. Diagramming the Universal Set (Sample Space) The universal set is usually diagrammed as a rectangle. The set name which is being used as the universal set is usually placed in the upper left hand corner of the shape. Depending on the size of the set you do not have to include all elements of the set in the diagram, usually a few are provided to give an image of some of the values of the set. If the set is small, then all elements should be listed. To diagram our sample space, the set M, a bag of marbles with 4 red marbles (solid) and 6 white marbles (empty) we create the rectangle, label it the universal set M, and then list out the elements of the set. In this case because there are only 10 elements it is easy to list them all out in the diagram. Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set) As stated earlier, a probability has two components, the sample space, which represents all possible things that could happen, and the defined successful outcomes, which represents the number of times a particular event occurs in that sample space. The outcome could be picking a heart from a deck of cards, rolling an even number on a dice, spinning a spinner and getting blue.. an outcome is simply a subset of the universal set. A subset is a collection of elements that all exist within another set. If all elements of set X belong to set Y, then it is said that set X is a subset of set Y. Any set formed with elements of the universal set is a subset of that universal set. For example if the sample space was rolling a D12 (a 12 sided dice) some subsets might be: Rolling a prime number, Set P = {2, 3, 5, 7, 11} is a subset of Rolling an even number less than 5, Set E = {2, 4} is a subset of Rolling a number greater than 12, Set B = {} is a subset of In the third example, Set B is an EMPTY SET or NULL SET. This means that no elements fit that description. The empty set gets its own special symbol, Ø. When notating an empty set we would write Set B = Ø, and NOT Set B = {Ø}. The latter notation is wrong because that set contains one element, the empty set. Thus set B would not be empty if it has one element, even if that element represents a set that has nothing. When writing that one set is a subset of another we use two special mathematical symbols, either or. The first symbol,, allows the subset to be the same as or smaller, whereas the second symbol,, forces the subset to contain less elements than the original set and these subsets are called proper subsets. Now if we look back to examples #1, #2 and #3, we would write those relationships as: Ex. #1 Set P Set U Ex. #2 Set E Set U Ex. #3 Set B Set U or Set U or Set P Set U or Set E Set U or Set B Set U or Set U
12.3 and 12.4 Notes Geometry 2 Diagramming a Outcome (Subset) using a Venn Diagram When a subset is defined, the elements are organized and a new boundary is drawn in the Venn diagram. So if we defined the set R as the set of all red marbles in the bag we would draw a new boundary that would contain all of those elements. Set R = {3R, 4R, 5R, 8R} Set U = {3R, 4R, 5R, 8R, 1W, 2W, 4W, 5W, 8W, 9W} Set R Set U This can easily be turned into a probability - What is the probability of picking a red marble from this bag of marbles? nr ( ) 4 P(Set R) = P(Red) = or 0.4 or 40% If we defined set E to be the set of all even numbers in the bag we could determine the probability to be: Set E = {4R, 8R, 2W, 4W, 8W} Set E Set U ne ( ) 5 P(Set E) = P(Evens) = or 0.5 or 50% Again if we defined set L to be the set of all numbers greater than 3 in the bag, we could determine the probability to be: Set L = {4R, 5R, 8R, 4W, 5W, 8W, 9W} Set L Set U nl ( ) 7 P(Set L) = P(Numbers >3) = or 0.7 or 70% The Complement of an Event, not The complement of an event is the probability of everything but that event occurring. So if the event was set A, then the complement is denoted as, set A c, everything that A is not. If the probability of picking a yellow marble from a bag is 3 8, then its complement, the probability of not yellow is. An easy way to 11 11 calculate the complement is P(A c ) = 1 P(A). This works because all probabilities sum to 1 and so whatever the probability of event A happening is, the probability of it not happening is everything else or in other words, 1 P(A). This relationship is easily viewed in a Venn diagram. P(A) + P(A c ) = 1
12.3 and 12.4 Notes Geometry 3 When determining the probability of a complement it is usually simplest to calculate the probability of the event and then subtract it from 1. Ex. #1 Given a bag of marbles with 3 green, 2 yellow and 5 red. What is the probability of NOT getting a green marble? P(G) = 3 10 1 P(G) = P(G c ) 1-3 10 = 7 10 Ex. #2 When rolling a single die, what is the probability of NOT getting a 6? P(A) = 1 6 1 P(A) = P(A c ) 1-1 6 = 5 6 Ex. #3 When picking a card from a standard deck, what is the probability of NOT getting a diamond? 1 P(A) = P(A c ) 1-13 52 = 39 52 Mutually Exclusive or Disjoint Sets More than one subset can be defined at a time from a universal set, so for example we could define the set of all red marbles, or the set of all even numbers, or the set of red marbles with numbers greater than 3 - the list seems like it could go on forever. Sometimes when we define more than one set at a time they have no elements in common. This is known as being mutually exclusive or disjoint. Two events are mutually exclusive events if the events cannot both occur in the same trial of an experiment, for example the flip of a coin cannot be both heads and tails and thus those two events are mutually exclusive. Diagramming Disjoint Sets If we define the set R to be all red marbles and the set W to be all white marbles we get two mutually exclusive sets because they have no elements in common with each other. We diagram this relationship by drawing boundaries around each set so that they do not touch or overlap in anyway. Set R = {3R, 4R, 5R, 8R} and Set W = {1W, 2W, 4W, 5W, 8W, 9W} Another example of disjoint sets would be set E, all of the even marbles, and set O, all of the odd marbles. Set E = {4R, 8R, 2W, 4W, 8W} and Set O = {3R, 5R, 1W, 5W, 9W} In both of these cases you cannot be both red and white or even and odd, thus they are mutually exclusive.
12.3 and 12.4 Notes Geometry 4 The Intersection, AND Of course when we define more than one subset the sets are not always mutually exclusive. Sometimes the two sets have shared or common elements in them. The shared items or elements are called the intersection of the sets. This should make sense to a Geometry or Algebra I student because we have already discussed the intersection of two lines. The intersection of two lines is a point, the only thing they HAVE IN COMMON. The Intersection The intersection is the collection of elements that are COMMON between the sets. The symbol notation for intersection is. In general, for any two sets S and T, the set consisting of the elements belonging to BOTH set S and set T is called the intersection of sets S and T, denoted by Set S Set T. This is sometimes also described as the elements that are in set S AND in set T. An example of two sets that would have an intersection could be found easily in a standard deck of cards, the set R, all red cards, and the set Q, the set of all queens. These two sets are NOT mutually exclusive because these sets would share two elements, the queen of hearts and the queen of diamonds. These two cards are the intersection because they are in set R AND in set Q. Another example of an intersection in a deck of cards would be the set D, the diamonds, and the set F, the face cards. The cards that are in set D AND set F (the intersection) are the jack, queen, and king of diamonds. Diagramming the Intersection If we define the set R to be all red marbles and the set E to be all even numbered marbles we get two sets that have an intersection. When these two set get diagrammed they have an overlapping region, a region that represents the values that are in both sets. We usually shade that region. Set R = {3R, 4R, 5R, 8R} and Set E = {4R, 8R, 2W, 4W, 8W} Set R Set E (Set R AND Set E) = {4R, 8R} Another example of an intersection would be the set D, all numbers divisible by 3, and the set W, all the white marbles. Set D = {3R, 9W} and Set W = {1W, 2W, 4W, 5W, 8W, 9W} Set D Set W (Set D AND Set W) = {9W} Could the intersection of two sets be empty? Of course if the two sets are mutually exclusive then there will be no elements in the intersection of the two sets. For example, the set E, the even numbered marbles and set O, the odd numbered marbles, will have no elements in common and so the intersection is the empty set. Set E = {4R, 8R, 2W, 4W, 8W} and Set O = {3R, 5R, 1W, 5W, 9W} Set E Set O (Set E AND Set O) =
12.3 and 12.4 Notes Geometry 5 The Union, OR The union of sets is exactly what it sounds to be, the process of combining sets together to form a larger set. The union of sets is the collection of all elements from both sets. The symbol for union is (this is easier to remember nion). In general, for any two sets S and T, the set consisting of all the elements belonging to at least one of the sets S and T is called the union of S and T, denoted Set S Set T. This is sometimes also described as the elements that are in set S OR in set T. An example of a union could be found easily in a standard deck of cards, the set R, all red cards, and the set S, the set of all spades. The union of these two sets would include all the hearts, all the diamonds and all the spades. These cards are the union because it contains set R OR set S. Diagramming the Union Usually we don t change the boundaries of the original sets to represent the new union; usually we simply shade in the sets that have formed the new union. The example to the right demonstrates the union of two mutually exclusive sets, set W, the white marbles {1W, 2W, 4W, 5W, 8W, 9W} and set E, the even red marbles {4R, 8R}. Set W Set E (Set W OR Set E) {1W, 2W, 4W, 5W, 8W, 9W} {4R, 8R} = {1W, 2W, 4W, 5W, 8W, 9W, 4R, 8R} An example of a union when the two sets that would have an intersection would be the Set E, the even numbers {2W, 4W, 8W, 4R, 8R} and the set R, the red marbles {3R, 4R, 5R, 8R}. Set E Set R (Set E OR Set R) {2W, 4W, 8W, 4R, 8R} {3R, 4R, 5R, 8R} = {2W, 4W, 8W, 3R, 4R, 5R, 8R} Let me do another example, the set B, the marbles greater than 2 and the set T, the marbles with a 3 or 4. Set B Set T (Set B OR Set T) {3R, 4R, 5R, 8R, 4W, 5W, 8W, 9W} {5R, 8R, 5W, 8W, 9W} = {3R, 4R, 5R, 8R, 4W, 5W, 8W, 9W} You do not double list elements in the set. You do not double list elements in the set.