PROBABILITY Vocabulary: Theory Book probability attributes Venn diagram interpret outcomes relationship mutually exclusive data values two-way table

What is probability? Probability is all about chance. What is the chance of something happening? For example: the weather report gives us a percentage chance that it will rain today. We can give a probability as a word or as a number The number can be written as a percentage, decimal or fraction

An event that is impossible is said to have a probability of 0 or 0% An event that is certain to happen is said to have a probability of 1 or 100% An event that is impossible is said to have a probability of 0 or 0% Any event in between impossible and certain have a probability in between 0 and 1 or between 0% and 100%. An even chance is 0.5 or 50% Fill in sheet with missing probabilities. Reiterate how to change from decimal to % to fraction where necessary. Could do as a class.

Some definitions in probability Theory Book Some definitions in probability Outcome – everything which can be a result of an event Sample Space – a list of all possible outcomes Exhaustive Event: Events that cover all possible outcomes Example: Rolling a number between 1 and 6 on a die This means rolling a 1,2,3,4,5 or 6. The sample space is {1,2,3,4,5,6} so an exhaustive event is the same as the sample space Hand out Sheets with terminology and definition on them and a set of cards to be placed in the spaces. Need to take in glue sticks. Discuss the words and what they mean and get students to stick definitions on the sheet.

Theory Book Writing Sample Spaces The Sample Space is the list of possible outcomes for a trial. Example: Rolling a die we can get any one of the numbers one to six. We write the sample space as {1,2,3,4,5,6} Example: Tossing a coin we can either get a head or a tail. We write the sample space as {H,T}

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Probability of an Event Occurring Theory Book Probability of an Event Occurring Probability = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑆𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑝𝑎𝑐𝑒 This means 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑤𝑒 𝑤𝑎𝑛𝑡 𝑐𝑎𝑛 ℎ𝑎𝑝𝑝𝑒𝑛 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Example: If I roll a die there are 6 possible numbers I can roll. This is the total number of possible outcomes or size of the sample space If I want to roll an even number there are 3 of them on a die (2,4,6) – This is 3 possible successful outcomes or 3 ways the event can happen P = 3 6 = 1 2 Your calculator can simplify it for you. Discuss and stick cards on bottom of the sheet. Students to write down example in their books

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Theory Book Complementary Events When we know the probability of something happening we also know the probability of it not happening (the opposite or complementary event) Example: Rolling a 4 on a normal die has P(4) = 1 6 The complementary event is ‘NOT rolling a 4’ P(not 4) = 5 6 The probabilities of an event and its complementary event add up to 1 P(4) + P(not 4) = 1 6 + 5 6 = 1

The probabilities of an event and its complementary event add up to 1 Theory Book The probabilities of an event and its complementary event add up to 1 P(4) + P(not 4) = 1 6 + 5 6 = 1 The probability of an event is written as P(E) The probability of a complementary event is P(E’) or P( 𝐸 ) We say P(E’) = 1 – P(E)

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Theory Book Compound Events Compound events are two or more events happening at the same time. Example: Rolling a 6 on a die and getting a tail on a coin. Calculating the probabilities associated with the two events is best done using a tree diagram or a table.

Tree Diagrams Theory Book Tree diagrams are used to list the sample space for experiments with two or more steps, and can make it easier to work out answers. The outcomes for each stage of the experiment are listed vertically and each stage is connected with branches. The tree below shows the possible outcomes when two coins are tossed. What is the chance of getting 2 heads? What is the chance of getting 2 tails? What is the chance of getting a head and a tail? Complete tree diagram. Then go through questions. Could go through more examples on the board if they don’t understand.

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Mutually Exclusive Events Theory Book Mutually Exclusive Events If one outcome can happen, then the other can’t; like ‘rolling an even number on a die’ and ‘rolling an odd number on a die’. Nothing can be BOTH of these.

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Theory Book Two way tables When we have two variables (things that change) we can show the options in a two way table and then calculate the probabilities of different events.

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Theory Book Venn Diagrams A Venn diagram is two or more circles that cross over for a section. We use Venn diagrams to show data points or people/items that fit into the categories for each circle If an item or person fits into more than one category they are placed in the overlap of the circles. Describe the diagram and where the numbers come from and what they mean. Venn Diagram activity outside in the quad or basketball courts.

Set Notation: Theory Book The sample space (list of all possible outcomes) is sometimes called the universal set and is given the symbol S, Ω, U or ξ. A is a particular subset (⊂) of the sample space if all the elements in A are contained in the sample space. For example, A is the set of prime numbers less than or equal to 10, which is a subset of all the integers less than or equal to 10. A′ is the complement of A and contains the elements not in A. 5 ∈ A means that 5 is an element of A (or is in set A). ∅ is the null or empty set and contains no elements. ∴ ∅ = { } n(A) is the cardinal number of A and means the number of elements in A. n(A) = 4

Theory Book A Venn diagram can be used to illustrate how different subsets in the sample space are grouped. For example: A = {2, 3, 5, 7} B = {1, 3, 5, 7, 9} A ∩ B means A and B, which means the intersection of A and B and includes the elements in common with both sets. ∴ A ∩ B = {3, 5, 7} A ∪ B means A or B, which means the union of A and B and includes the elements in either A or B or both. ∴ A ∪ B = {1, 2, 3, 5, 7, 9} A only is the elements in A but not in B. ∴ A only = {2}

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Theory Book Relationships: Information can be recorded both using a Venn diagram and a two-way table:

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