probability for beginners

Probability for Beginners

Concept 1: Sample Space as a Set

• Probability is often defined as how likely something is to happen.

• To determine how likely something is to happen we must 1st find the total number of outcomes that could happen for any given event or experiment.

• The list of all possible outcomes is called the sample space. – Sample space is denoted: S= { , , , , …}

Concept 1: Sample Space as a Set • A sample space has

uniform probability if every outcome is equally likely to happen.

• Which of these have uniform probability?


• Some sample spaces are too large to list, like a deck of cards.

• Instead of writing the sample space in set notation, we can also create a tree diagram!


• Create a tree diagram showing all possible outcomes (the sample space) of flipping a coin twice.



TOTAL(2)(2) = 4

2ND FLIP (2)

1ST FLIP (2)

TT

TH

HT

HH

TAIL

TAIL

TAIL

HEAD

HEAD

HEAD



TOTAL(2)(2) = 4

2ND FLIP (2)

1ST FLIP (2)

TT

TH

HT

HH

TAIL

TAIL

TAIL

HEAD

HEAD

HEADThis is our sample space. Is it uniform or not uniform?

Concept 1: Sample Space as a Set • Try another one. This time, create a tree diagram depicting the sample

space for “choosing 2 scoops of ice cream” given the following 3 flavors:

This is our sample space. Is it uniform or not uniform?

(3)(3) = 9OPTIONS

3 OPTIONS3 OPTIONS

SS

SC

SV

CS

CC

CV

VS

VC

VV

VANILLA

CHOCOLATE

STRAWBERRY

VANILLA

CHOCOLATE

STRAWBERRY

STRAWBERRY

CHOCOLATE

STRAWBERRY

CHOCOLATE

VANILLA

VANILLA


• Lastly, we can also use a table to show a sample space. For example, the following table depicts the sample space for rolling a die twice.

This is our sample space. Is it uniform or not uniform?

Concept 1: Sample Space as a Set • When dealing with the occurrence of more than one event or activity

like rolling a die 2 times, it is important to be able to quickly determine how many possible outcomes exist without having to list 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.

Concept 1: Sample Space as a Set • Use the Fundamental Counting Principle to determine the total number of

possible outcomes for each of the following events.

Concept 2: Diagramming the Sample Space & Outcomes Using Venn Diagrams


• As stated earlier, a probability has two components: – the sample space (all possible things that could happen) – the defined successful outcomes, aka an “event” (the # of times a

particular outcome occurs in the 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 event is simply a subset of the sample space that may contain 1 or more outcomes. – 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.


Set B is an EMPTY 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


When writing that one set is a subset of another we use two special mathematical symbols, either or . means the subset is less than or equal to the original set. means the subset is only less than the original set. (These subsets are called proper subsets.) Describe the relationships between each of the sets above: Set P Set U Set E Set U Set B Set U or Ø Set U

⊆⊆

⊂⊂


• When a subset of the sample space 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.

P(Set R) = P(Reds) = _____


• So if we defined the set E as the set of all even # marbles in the bag we would draw a new boundary that would contain all of those elements.

P(Set E) = P(Evens) = _____

Concept 3: 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, its complement is denoted as, set Ac,

“everything that A is not”. – Example: If the probability of picking a yellow marble from a bag is 3/11

, then its complement, the probability of not yellow is _______.

• An easy way to calculate the complement is P(Ac) = 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(Ac) = 1 So

P(Ac) = 1 – P(A)


• 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?


Ex. #2 – When picking a card from a standard deck, what is the probability of NOT getting a diamond?

Concept 4: Mutually Exclusive or Disjoint Events

• More than one subset can be defined at a time from a sample space – For example we could define

• the set of all red marbles • the set of all even numbers • the set of red marbles with numbers greater than 3

• 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 1 flip of a coin cannot be both heads and tails.

Concept 4: Mutually Exclusive or Disjoint Events

In both of these cases you cannot be both red and white or even and odd, thus they are mutually exclusive.

Concept 5: Intersection, “AND”

• Sometimes the two sets have shared or common elements in them. The shared items or elements are called the intersection of the sets.

• The intersection is the collection of elements that are COMMON between both 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.

∩

∩


Question: Could the intersection of two sets be empty?

∩


∩

Concept 6: 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 . • 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.

∪

∪

Concept 6: Union, “OR”

Your Assignments:

• “Beginner Probability Wkst”

probability for beginners

Documents