HSS-CP.A.1 STUDENT NOTES WS #1 – geometrycommoncore.com 1 Sample Spaces as Sets

Probability is often defined as how likely something is to happen. In determining how likely something is to happen we must first determine what the total number of things that could happen for any given event or experiment. The list of all possible outcomes is called the sample space.

A LIST -- A sample space is often organized into a set of elements. For example the sample space for rolling a die is {1, 2, 3, 4, 5, 6} or the sample space for flipping a coin is {Head, Tail}. These two sample spaces represent uniform probability. Uniform probability is where each element of the set has the same chance of happening or in other words, each element is equally likely to happen. In a uniform probability, the number of elements in the set is the total number of outcomes possible for the sample space.

Some sample spaces are not uniform, such as a bag of marbles with 2 red and 1 green, the sample space is {red, green} because only a red or a green marble can be chosen. But in this sample space each element is NOT equally likely to happen because there are 2 ways to pick a red marble and only 1 way to pick a green. While there are only two elements listed in the sample space, the total number of outcomes of this sample space is 3, written n(S) = 3, because red has 2 ways of obtaining it and green has 1. The notation n(S) refers to the total number of outcomes possible in the set. Also notice when we list the elements of a set we use the set brackets {} and each element in the set is separated by a comma. {12, 34} is a set containing only two elements, whereas the set {1, 2, 3, 4} is a set containing 4 elements.

Some typical sample spaces written in set notation are:

Flipping a Coin Rolling a Six Sided Dice Spinning a 4 color spinner Rolling a Weighted Dice

SAMPLE SPACE SAMPLE SPACE SAMPLE SPACE SAMPLE SPACE {Head, Tail} {1, 2, 3, 4, 5, 6} {Red, Yellow, Green, Blue} {4, 5, 6} Uniform Uniform Uniform Not Uniform Picking a flavor of ice Determining the gender Picking from a bag of marbles cream of baby

SAMPLE SPACE SAMPLE SPACE SAMPLE SPACE {Chocolate, Vanilla, {Boy, Girl} (Blue, Red} Strawberry} Uniform Uniform Not Uniform HSS-CP.A.1 STUDENT NOTES WS #1 – geometrycommoncore.com 2

Some sample spaces can be quite large and difficult to list such as the 52 cards in a standard deck.

Large sample spaces can make listing them a daunting task and often quite inefficient. For this reason we have a few other ways to organize a sample space.

A TREE DIAGRAM. The first tree diagram below shows two flips of a coin. The first flip could be a head or a tail and then the second flip could produce a head or a tail from each of the previous options. The tree displays the sample space for flipping a coin twice as {HH, HT, TH, TT}. We also notice the number of elements of the sample space is (2)(2) = 4. In the second tree diagram below we see the sample space for selecting two scoops of ice cream where the choices are vanilla, chocolate, and strawberry. There are three choices for the first scoop and then the second choice gives us three more options for each previous option. The tree diagram displays the sample space for choosing a double scoop as {VV, VC, VS, CV, CC, CS, SV, SC, SS}. Again we notice an easy method for determining the number of elements in this sample space is to multiply 3 by 3 to get 9.

Tree Diagram for Flipping a Coin Twice Randomly Selecting a Two Scoops Ice Cream Cone

HEAD HH VANILLA VV

HEAD VANILLA CHOCOLATE VC TAIL HT STRAWBERRY VS VANILLA CV HEAD TH CHOCOLATE CHOCOLATE CC TAIL STRAWBERRY CS TAIL TT VANILLA SV 1ST FLIP 2ND FLIP TOTAL (2) (2) (2)(2) = 4 STRAWBERRY CHOCOLATE SC

STRAWBERRY SS

3 OPTIONS 3 OPTIONS (3)(3) = 9 OPTIONS

In this case, getting a head is as likely as In this case, the choice of flavors might not be equally likely. getting a tail. Thus this tree diagram If chocolate was the most popular flavor it would have a HSS-CP.A.1 STUDENT NOTES WS #1 – geometrycommoncore.com 3 represents a uniform probability. In this greater chance of being selected. The fact that each choice sample space each event is equally likely – would not be equally likely would make the probabilities getting a {Tail, Tail} is as likely as getting a non-uniform. For example, getting {Chocolate, Chocolate} {Head, Tail} or any of the other events. would be more likely to occur than {Strawberry, Strawberry}.

A TABLE OR CHART. A tree diagram would probably not be the best way to organize the sample space for all possible outcomes when rolling two dice. The tree would have 6 initial branches and then for each of those branches there would 6 more branches – in just two levels of the tree we would already have 36 branches. This would create a very large (hard to draw) tree diagram. A better way to organize this data might be to create a table.

1 2 3 4 5 6 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

When dealing with the occurrence of more than one event or activity such as this one, it is important to be able to quickly determine how many possible outcomes exist without listing all of the possible events. In this case to determine the total number of outcomes we could simply multiply 6 times 6 to get 36. This simple multiplication process is known as the Fundamental Counting Principle.