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4.3 Probability Using Set Notation TERMINOLOGY Venn Diagram ≡ A diagram in which sets are represented by shaded or coloured geometrical shapes Compound Event ≡ Consists of two or more simple events Subset ≡ A set whose members (values) are all members of another set 4.3 Probability Using Set Notation Intersection of Sets ≡ Elements, members, or values that are common to both (all) sets that intersect A ∩ B = {elements in both A and B} Set of common elements to A and B Ex. – Some students take French and some take English while some take both French & English. English French French & English 4.3 Probability Using Set Notation Union of Sets ≡ Set formed by joining or combining all elements from set A and all elements from set B A U B = {elements in A or B} Set formed by combining elements in A or B Ex. – Some students take French and some take English while some take both French & English. What does the shaded region represent? 4.3 Probability Using Set Notation Disjoint Sets ≡ occur when sets have no elements in common and their intersection is the empty set (Φ ) n(A∩B)=Φ Ex. Some students take English and some take French. No one takes both A ∩ B = Φ These sets are A.K.A. Mutually Exclusive events 4.3 Probability Using Set Notation Additive Principle for Unions of Two Sets To count the number of elements in two groups or sets that have common elements, you must subtract the common elements so that they are not included twice For sets A and B, the total number of elements in the union of A & B is the number in A plus the number in B minus the number in both A and B n(A or B)=n(A)+n(B)-n(A and B) n(A U B)=n(A)+n(B)-n(A ∩ B) 4.3 Probability Using Set Notation Somethings to think about! Why must we subtract n(A ∩ B) in the general formula? Under what circumstances would n(A U B)=n(A) + n (B)? 4.3 Probability Using Set Notation Using Venn Diagrams to Solve Counting Problems Example Of the 140 grade 12 students at SFBSS, 52 have signed up for biology, 71 for chemistry, and 40 for physics. The science students include 15 taking both biology and chemistry, 8 taking chemistry and physics, 11 taking biology and physics, and 2 who are taking all three. How many are not taking any science courses? Illustrate the enrolments with a Venn Diagram. 4.3 Probability Using Set Notation Extended Additive Principle B≡Biology, C≡Chemisty, P≡Physics n(total)=n(B)+n(C)+n(P)-n(B ∩ C)-n(C ∩ P)-n(B ∩ P)+n(B ∩ C ∩ P) =52+71+40-15-8-11+2 =131 Therefore, there are 131 students taking one or more science courses & (140-131) 9 students not taking science in Gr 12 4.3 Probability Using Set Notation Venn Diagram Start with overlap amongst 3 courses and work outward. B C 28 13 50 2 9 6 23 P 4.3 Probability Using Set Notation Additive Principle for Unions of 2 Sets n(A U B)=n(A) + n (B) – n(A ∩ B) Total number of elements in both sets then subtract the number that have been counted twice 4.3 Probability Using Set Notation Probability Using Additive Principle: Probability of the Union of Two Events P(A U B) = P(A) + P(B) – P(A ∩ B) If A and B have no outcomes in common (are disjoint), A and B are said to be mutually exclusive events and therefore – P(A U B) = P(A) + P(B) {since P(A ∩ B)=0} 4.3 Probability Using Set Notation 4.3 Probability Using Set Notation Extend our SFBSS science student to probability. Determine the probability that a student selected at random from the group is taking Biology. n( B) 52 13 P( B) 37 % n( S ) 140 35 4.3 Probability Using Set Notation Determine the probability that a student selected at random from the group is taking Chemistry or Physics. P (C P ) P (C ) P ( P ) P (C P ) 71 40 8 140 140 140 103 140 73.6% 4.3 Probability Using Set Notation Example: Consider the sets A={5,6,7,8} and B={3,4,7,10} What is A ∩ B? What is A U B? Make set C that is disjoint to B. 4.3 Probability Using Set Notation Example Create a Venn diagram for the following scenerio. 1500 students attend a local high school. The first school dance had 740 students in attendance, while only 440 attended the second dance. If 285 students attended both, how many did not go to either dance? (find n(A U B) and then its compliment) Answer = 605 4.3 Probability Using Set Notation Example The probability that Sarah will get into university is 2/5 and the probability that she will get into college is 3/4. If the probability that she gets into both is 1/5, what is the probability that she will get into at least one of the schools? Answer=19/20 or 95% 4.3 Probability Using Set Notation Example An advertiser is told that 70% of all adults in the GTA read the Toronto Star and 60% watch CityTV. She is also told that 90% either read the Star or watch CityTV. If she places an ad in the paper and runs a commercial, what is the probability that an adult selected at random in the GTA will see both of these ads? Answer = 0.4 or 40% 4.3 Probability Using Set Notation Home Entertainment Pg 228 #1-15

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posted: | 4/21/2011 |

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