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5-Minute Check on Activity 6-2

5-Minute Check on Activity 6-2. How many different meals can be served with 3 appetizers, 5 entries, and 4 deserts? Finish filling out the following tree diagram: If P(event) = 0.52, what is the probability of its complement? If P(no one passing) = 0.13, what is the P(at least one passes)?.

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5-Minute Check on Activity 6-2

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  1. 5-Minute Check on Activity 6-2 • How many different meals can be served with 3 appetizers, 5 entries, and 4 deserts? • Finish filling out the following tree diagram: • If P(event) = 0.52, what is the probability of its complement? • If P(no one passing) = 0.13, what is the P(at least one passes)? 3×5×4 = 60 different meals 0.7 0.42 0.18 0.28 0.12 ?? ?? ?? ?? y.y YY Question 2 0.6 YN 0.3 Question 1 0.4 0.7 y.y NY x.x Question 2 0.3 NN P(complement) = 1 – P(event) = 1 – 0.52 = 0.48 P(at least one) = 1 – P(none) = 1 – 0.13 = 0.87 Click the mouse button or press the Space Bar to display the answers.

  2. Activity 6 - 3 Experimenting With Probabilities

  3. Objectives • Distinguish between independent and dependent events • Calculate probability problems involving “and” statements using multiplication formula • Calculate probability problems involving “or” statements using addition formula • Identify mutually exclusive events

  4. Vocabulary • Independent – events are independent if the occurrence of either event does not affect the probability of the other occurring • Dependent– events are dependent if the occurrence of either event affects the probability of the other occurring • Mutually Exclusive – two events are mutually exclusive if the both cannot occur at the same time

  5. Activity Consider the following two probability experiments. Experiment1: You roll two fair dice. What is the probability that a 4 show on each die? 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 that the cards are both red suited? These experiments are very similar. In each case, you are determining the probability that two events occur together. The word and is used to indicate that both events must occur. In Venn diagrams the word and is generally interpreted to mean the intersection of two sets.

  6. 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

  7. 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.

  8. Independent vs Dependent • Events A and B are independent if the occurrence of either event does notaffect the probability of the occurrence of the other event • Events A and B are dependent if the occurrence of either event doesaffect the probability of the occurrence of the other event

  9. 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

  10. 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

  11. Venn Diagrams in Probability • A  B is read A union B and is both events combinedalso 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 A B B Mutually Exclusive Intersection: A  B

  12. 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, thenP(E and F and G and …..) = P(E) ∙ P(F) ∙ P(G) ∙ ……

  13. 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)

  14. 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

  15. 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

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

  17. 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

  18. 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) • Homework • pg 731-737; problems 1, 2, 4, 8, 13

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