Good afternoon! August 9, 2017

Bellringer 1. 4.25 x 5.2 = 2. 1.2 x 3.5 =

Concept Check ___Probability___ ___Venn Diagram___ ___Complement___ ___Independent___ ___Addition Rule___

Standard: MGSE9-12.S.CP.2 Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent. Learning Target: I CAN represent real world objects algebraically. I CAN communicate mathematically using set notation. I CAN make everyday decisions by understanding conditioned probability.

Experimenting With Probabilities

Vocabulary Independent – event does not affect the 2nd event Dependent – event does affect the 2nd event Mutually Exclusive – two events are mutually exclusive if they both cannot occur at the same time

Activity – Experiment 1 You roll two fair six-sided dice. What is the probability that a 4 is rolled with the first die?   What is the probability that a 4 was rolled on the second die? What is the probability that a 4 show on each die? P(4) = 1/6 = 0.166 P(4) = 1/6 = 0.166 P1(4)  P2(4) = 1/6  1/6 = 1/36 = .0278

Activity – Experiment 2 You pick two cards from a standard deck of 52 cards without replacing the first card before picking the second. What is the probability of a red suited card on the first draw?   What is the probability of a red suited card on the second draw, given that we got one on the first draw?  What is the probability that the cards are both red suited? Why are these two experiments different? P(RS) = 26/52 = 0.5 P2(RS) = 25/51 = 0.4902 P(both RS) = P1(RS)  P2(RS) = 0.5  0.4902 = 0.2451 In the first experiment, events are independent. Not so in the second – events are dependent.

Independent vs Dependent Events A and B are independent if the occurrence of either event does not affect the probability of the occurrence of the other event Events A and B are dependent if the occurrence of either event does affect the probability of the occurrence of the other event

Independent Examples What is the probability of rolling 2 consecutive sixes on a ten-sided die? What is the probability of flipping a coin and getting 3 tails in consecutive flips? P(6 and 6) = P(6)  P(6) = 1/10  1/10 = 1/(102) = 1/100 = 0.01 P(H and H and H) = P(H)  P(H)  P(H) = 1/2  1/2  1/2 = 1/(23) = 1/8 = 0.125

Dependent Examples What is the probability of drawing two aces from a standard 52-card deck? What is the probability of drawing two hearts from a standard 52-card deck? P(A1 and A2) = P(A1)  P(A2) = 4/52  3/51 = 12/2652 = 0.005 P(H1 and H2) = P(H1)  P(H2) = 12/52  11/51 = 132/2652 = 0.050

Venn Diagrams in Probability A  B is read A union B and is both events combined also seen as A or B A  B is read A intersection B and is the outcomes they have in common also seen as A and B Disjoint events have no outcomes in common and are also called mutually exclusive In set notation: A  B =  (empty set) A B A B Mutually Exclusive Intersection: A  B

Multiplication Rule: Independent Events If A and B are independent events, then P(A and B) = P(A) ∙ P(B) If events E, F, G, ….. are independent, then P(E and F and G and …..) = P(E) ∙ P(F) ∙ P(G) ∙ ……

Addition Rule: Disjoint Events If E and F are disjoint (mutually exclusive) events, then P(E or F) = P(E) + P(F) E F Probability for Disjoint Events P(E or F) = P(E) + P(F)

Addition Rule for Disjoint Events If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(A or B or C) = P(A) + P(B) + P(C) This rule extends to any number of disjoint events

Mutually Exclusive Examples What is the probability of rolling a 6 or an odd-number on a six-sided die? What is the probability of drawing a king, queen or jack (a face card) from a standard 52-card deck? P(6 or odd) = P(6) + P(odd) = 1/6 + 3/6 = 4/6 = 2/3 = 0.667 P(K or Q or J) = P(K) + P(Q) + P(J) = 4/52 + 4/52 + 4/52 = 12/52 = 3/13 = 0.231

General Addition Rule For any two events E and F, P(E or F) = P(E) + P(F) – P(E and F) E F E and F Probability for non-Disjoint Events P(E or F) = P(E) + P(F) – P(E and F)

General Addition Examples What is the probability of rolling a 5 or an odd-number on a six-sided die? What is the probability of drawing a red card or a face card from a standard 52-card deck? P(5 or odd) = P(5) + P(odd) – P(5 and odd) = 1/6 + 3/6 = 4/6 = 2/3 = 0.667 P(R or FC) = P(Red) + P(FC) – P(RFC) = 12/52 + 26/52 – ( 6/52 ) = 32/52 = 0.6154

Summary and Homework Summary Events are independent if the occurrence of either event does not affect the occurrence of the other P(A and B) = P(A)  P(B) Events are dependent if the occurrence of either event does affect the occurrence of the other P(A and B) = P(A)  P(B|A) (remember for next lesson) Mutually exclusive events can’t happen at same time P(A or B) = P(A) + P(B) (P(A and B) = 0) Events not mutually exclusive P(A or B) = P(A) + P(B) – P(A and B)