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Key ideas
Key ideas
I can construct and use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
I can determine and explain the difference between events with replacement and without replacement
I can explain and use combined events to solve problems: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
I can explain and use mutually exclusive events to solve problems: P(A ∩ B) = 0
I can calculate problems involving conditional probability: P(A ∩ B) = P(A|B) / P(B)
I can explain what independent events are: P(A ∩ B) = P(A)P(B)
Sets and Venn diagrams
Sets and Venn diagrams
This section will be new to some students. It takes probability in a new direction. Make sure to study this one thoroughly.
Venn Diagrams - An Introduction. In this video, I give an intro to Venn Diagrams by labeling a Venn Diagram with numerical values using a concrete example. In other videos, I will discuss the union/intersection notation.
Regions with two sets
Regions with two sets
Venn diagrams are another way of presenting probability information. We can visualise the outcomes by shading areas. There is also some new language to be learnt: Symbols that represent AND (the intersection of sets), OR (the union of sets) etc.
https://www.youtube.com/watch?v=b6t0994ZZDA
Venn Diagrams: Shading Regions. In this video I show how to shade the union, intersection and complement of two sets.
Venn diagram calculator.
Venn diagram calculator.
A simple tool to help place numbers in a Venn diagram.
https://www.geogebra.org/material/show/id/AjrfISIT
Probability Laws
Probability Laws
P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Note: if A and B are mutually exclusive: P(A ∩ B) = 0
Conditional Probability
Conditional Probability
P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A).
Rearranged this gives the more useable: P(A|B) = P(A ∩ B) ÷ P(B)
Mutually Exclusive events and Conditional Probability
Mutually Exclusive events and Conditional Probability
Conditional Probability is where the outcome of an experiment depends on the outcome of another (previous) experiment. Using Venn diagrams we can easily visualise the laws of probability. These laws will allow us to work with probability in an algebraic way.
Mutually exclusive events can not occur at the same time!
Experiments with Replacement
Experiments with Replacement
All the experiments are the same in this case and therefore have the same probability e.g.
Flipping a coin n times
Rolling a dice n times
Selecting an M&M from a bag and then putting it back in.
Experiments without Replacement
Experiments without Replacement
The successive experiments do not have the same probability e.g.
Selecting an M&M from a bag and then eating it (the total number of M&Ms has now changed!).
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