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Section 12.7 Probability of Compound Events
Day 1 - Algebra 1
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Probability = π‘ππ‘ππ # ππ πππ£ππππππ ππ’π‘πππππ π‘ππ‘ππ # ππ ππ’π‘πππππ
Recall Probability = π‘ππ‘ππ # ππ πππ£ππππππ ππ’π‘πππππ π‘ππ‘ππ # ππ ππ’π‘πππππ
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Activity Part 1 Determine all of the sums that are generated when rolling 2 dice. Answer: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
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Activity Part 1 Make a prediction! Do all of the sums have an equal chance of occurring, or will some sums have a larger chance of occurring?
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Activity Part 2 Random Generator βMATHβ βPRBβ βrandInt(β
Sum of Dice # of Times 2 3 4 5 6 7 8 9 10 11 12 Part 2 40 times per pair, then total table Random Generator βMATHβ βPRBβ βrandInt(β randInt(1, 6, 2) Enter
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Activity Part 3 Compile Results! Experimental Probability Sum of Dice
# of Times 2 3 4 5 6 7 8 9 10 11 12 Part 3 Compile Results! Experimental Probability
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Activity Part 3 Compile Results! Theoretical Probability Sum
Possible Pairs 2 1+1 3 1+2, 2+1 4 1+3, 2+2, 3+1 5 1+4, 2+3, 3+2, 4+1 6 1+5, 2+4, 3+3, 4+3, 5+1 7 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 8 2+6, 3+5, 4+4, 5+3, 6+2 9 3+6, 4+5, 5+4, 6+3 10 4+6, 5+5, 6+4 11 5+6, 6+5 12 6+6
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Independent Events Definition: outcome of one event does not affect the outcome of the other event Example: The plane being on time may not affect whether the luggage is lost.
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Probability of Independent Events
π π΄ πππ π΅ =π(π΄)βπ(π΅) Event A and event B are independent events. The probability of both events occurring is the product of each individual probability.
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Example 1 Rae is flying from Birmingham to Chicago on a flight with a 90% on-time record. On the same day, the chances of rain in Denver are predicted to be 50%. What is the probability that Raeβs flight will be on time and that it will rain in Denver? Answer: π π΄ πππ π΅ =π(π΄)βπ(π΅) π ππ π‘πππ & ππππ =π(ππ π‘πππ)βπ(ππππ) = 0.9 β 0.5 =0.45 The probability that Raeβs flight will be on time and that it will rain in Denver is 45%.
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Example 2 Two cities, Fairfield and Madison, lie on different faults. There is a 60% chance that Fairfield will experience an earthquake by the year 2020 and a 40% chance that Madison will experience an earthquake by Find the probability that both cities will experience an earthquake by 2020. Answer: 0.4 β 0.6 =0.24=24%
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Dependent Events Definition: outcome of one event affects the outcome of the other event Example: A marble is selected from a bag, but not replaced. Then, a second marble is selected from the bag.
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Probability of Dependent Events
π π΄ πππ π΅ =π(π΄)βπ(π΅ ππππππ€πππ π΄) Event A and event B are dependent events.
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Example 3 At the school carnival, winners in the ring-toss game are randomly given a prize from a bag that contains 4 sunglasses, 6 hairbrushes, and 5 key chains. Three prizes are randomly drawn from the bag and not replaced. Find π(π π’πππππ π ππ , βππππππ’π β, πππ¦ πβπππ). Answer: π π π’πππππ π ππ = 4 15 π βππππππ’π β = 6 14 π πππ¦ πβπππ = 5 13 Multiple all three together = 4.4%
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Example 5 A gumball machine contains 16 red, 10 blue, and 18 green gumballs. Once a gumball is removed from the machine, it is not replaced. Find the probability if the gumballs are removed in the order indicated π πππππ, πππ’π, πππ πππ . Answer: π(πΊππππ)= π π΅ππ’π = π πππ πππ = Multiple all three together = 5.9%
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Mutually Exclusive Events
Definition: Events that cannot occur at the same time. Example: Finding the probability of drawing a heart or a diamond. A card cannot be a heart and a diamond at the same time.
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Mutually Exclusive Events Probability
π π΄ ππ π΅ =π π΄ +π(π΅) Event A and event B are mutually exclusive events.
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Example 6 A card is being drawn from a standard deck. Find the probability of π 7 ππ 8 . Answer: π(7)= 4 52 π 8 = 4 52 Add together: 8 52 =15.4%
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Example 7 A card is being drawn from a standard deck. Find the probability of π(ππππ‘βππ πππ’π πππ βππππ‘). Answer: π(πΆππ’π)= 13 52 π π»ππππ‘ = 13 52 Add together: (this is the probability of getting a club or a heart. To NOT draw a club or heart, subtract from 1.) 1β = 1 2 =50%
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Not Mutually Exclusive Events
Definition: Events that can occur at the same time. Example: Finding the probability of drawing a red card or an even card.
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Not Mutually Exclusive Events Probability
π π΄ ππ π΅ =π π΄ +π π΅ βπ(π΄ πππ π΅) Event A and event B are not mutually exclusive events.
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Example 8 In Mrs. Klineβs class, 7 boys have brown eyes and 5 boys have blue eyes. Out of the girls, 6 have brown eyes and 8 have blue eyes. If a student is chosen at random from the class, what is the probability that the students will be a boy or have brown eyes. Answer: π π΄ = (boy) π π΅ = (brown eyes) π π΄ πππ π΅ = 7 26 (boy & brown eyes) π π΄ ππ π΅ = β 7 26 = 9 13
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Example 9 Of 240 girls, 176 are on the Honor Roll, 48 play sports, and 36 are on the Honor Roll and play sports. What is the probability that a randomly selected student plays sports or is on the Honor Roll? Answer: π π΄ = (sports) π π΅ = (Honor Roll) π π΄ πππ π΅ = (sports and honor roll) π π΄ ππ π΅ = β = 47 60
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Identify Independent, Dependent, Mutually Exclusive, Not Mutually Exclusive A) Picking a blue, then a red marble from a bag with replacement. B) The probability that a household has a dog or a cat. C) A die is rolled. The face of the die being 3 or 5. D) Choosing 3 cards from a deck without replacement Answers: A) Independent, B) Not Mutually Exclusive, C) Mutually Exclusive, D) Dependent
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Homework Page #10, 11, 15, 16, 19, 24
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