How likely something is to happen - Ms. Hale's Website

[Pages:6]Notes: Probability for Beginners

Name_________________________________________ P______

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 ______________________ of outcomes that could happen for any given event or experiment.

? The list of all possible outcomes is called the __________________________________________________.

Sample space is denoted:

?A sample space has _____________________________________________ 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, so instead of writing it in set notation, we can also create a ___________________________________________.

?Create a tree diagram depicting the sample space for "choosing 2 scoops of ice cream" given the following 3 flavors:

?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.

?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 ___________________________ to get ____________. This simple multiplication process is known as the Fundamental Counting Principle.

?Use the Fundamental Counting Principle to determine the total number of possible outcomes for each of the following events.

C

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

?As stated earlier, a probability has two components:

? the _________________________________________ (all possible things that could happen)

?the defined ___________________________________________, aka an "_______________________" (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_______________________ is simply a _________________________ of the sample space that may contain _____________ or _______________ outcomes.

?A ___________________________ 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.

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 ?

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.

?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.

Concept 3: The Complement of an Event, "NOT" ?The complement of an event is the probability of _______________________________________________________________.

? So if the event was set A, its complement is denoted as, ________________, "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) =____________________________________ ?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.

?When determining the probability of a complement it is usually simplest to calculate the _________________________________________________________________and then ______________________________________________. 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 ?Sometimes when we define more than one set at a time they have _______________elements in common. This is known as being _____________________________________________ or ___________________________. ?Two events are mutually exclusive events if the events ______________________ both occur in the same trial of an experiment

?For example 1 flip of a coin cannot be both _______________ and _________________.

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 ____________________________________ of the sets. ?The ______________________________ is the collection of elements that are ____________________ 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 __________________________________.

? This is sometimes also described as the elements that are in set S __________ in set T.

Concept 6: Union, "OR"

?The union of sets is exactly what it sounds to be, the process of _______________________________________________ to form a larger set. The union of sets is the collection of __________________________________________ from ____________________________________.

?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 _________________________________.

?This is sometimes also described as the elements that are in set S ________ in set T.

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