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Conditional Probability in Venn Diagrams Slideshow 59, Mathematics Mr Richard Sasaki, Room 307 Objectives • Learn to draw Venn Diagrams for Four Sets • Review and picture sections in simple Venn Diagrams (two sets) • Understand Conditional Probability in Venn Diagrams 4 Set Venn Diagrams A regular Venn Diagram must cover all options. It cannot be made simply with just circles. This pattern does not cover Red and Green or Yellow and Blue. In fact, to cover all possibilities, there must be 16 sections. (including the outside). 4 Set Venn Diagrams Some solutions are… This one has 11 sets… (It was discovered 3 years ago.) What makes this special is that it’s symmetrical. Introduction We know how to use Venn Diagrams now (this is our third slideshow on Venn Diagrams). Set A ∩ ′ Set B ∩ ′ ∩ The total number of values in this diagram is the number of entries surveyed. ′ ∩ ′ Number of Entries = ∩ ′ + ∩ + ′ ∩ + ′ ∩ ′ ∩ ′ + ∩ + ′ ∩ + ′ ∩ ′ = 1 Probabilities If we choose one of the entries surveyed at random, we can find the probability of it being in its given category. Example 11 Set A 13 Set B 7 19 Total # of entries: 50 20 = 50 24 ′ = 50 ( ∩ ) = ( ∩ ′) = 13 50 (′ ∩ ′) = 11 50 7 50 ( ∪ ) = Note: ∪ = 1 − (′ ∩ ′). 39 50 Shading Exercise Firstly I’d like us to do a shading in exercise. Shade in the areas mentioned on your worksheet. Answers Conditional Probability Conditional probability refers to… The likelihood of something happening changing, depending on whether something else happens or not. (For example: If A happens, P(B) is lower than when A doesn’t happen.) When in daily life might this happen? Wearing Glasses and taking an eye exam Passing an exam after studying for hours Being able to kiss a girl after eating garlic Conditional Probability Note: Most of your examples would have been two events happening in turn (like with tree diagrams). But Venn diagrams show relationships as a whole, ignoring time. For this we use given that which has the ‘|’ symbol. What does P(|) mean? It means, assuming A is true, find the chance of B being 2 Set A Set B true. 4 3 5 How about for the Venn diagram on the right? = 3 7 which is different from =8 14. Conditional Probability Example A group of 50 people are asked whether they wear glasses and whether they wear contact lenses. 23 people wear glasses (Set A), 18 wear contacts (Set B) and 14 use both. 23 Set A 9 Set B 14 4 Total # of entries: 50 If picked randomly, find… 23 = 50 | = | = |′ = Try the ActivExpression exercise! 14 23 14 18 9 32 6 14 40 40 5 5 26 13 8 13 100 30 ∩ = 100 29 | = 59 ′ (A ∪ )= (A′|′)= Set A Set B 30 29 13 72 100 28 58 28 Set A Set C Set B