Mutually Exclusive and Independent Events Intersection: A ∩ B A AND B Overlap Only! Union: A ∪ B A OR B Anything in A, B, or Both Probability of A OR B (Union): P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Mutually Exclusive: If the intersection is empty and thus P(A ∩ B) = 0 Probability of A AND B (Intersection): P(A ∩ B) = P(A) * P(B) Independent: Occurrence of one has no impact on occurrence of the other. Independent only if P(A ∩ B) = P(A) * P(B) Example 3 - Finding Probabilities of Complements When two six-sided dice are rolled, there are 36 possible outcomes, as shown. a. The sum is not 5 P(S ≠ 5) = 1 - P(S = 5) = 8/9 b. The sum is less than or equal to 8 P(S ≤ 8) = 1 - P(S > 8) = 13/18 Example 4 - Using a Complement in Real Life a. There are 10 people at a dinner party. What is the probability that at least 2 people have the same birthday? P(2 or more) = 1 - P(None) = 1 - (365/365) * (364/365) * ... = 1 - (365! / 365^10) ≈ 0.1169 b. A restaurant gives a free different message hidden inside the fortune cookie to every guest. The restaurant claims there are 100 different messages hidden inside the fortune cookies. What is the probability that a group of 5 people receive at least 2 fortune cookies with the same message inside? P(2 or more) = 1 - (100/100) * (99/100) * ... = 1 - (100P5 / 100^5) ≈ 0.0415 EXAMPLE 5 - Finding Probabilities of Independent Events a. In a survey, 50 of 75 male fans and 40 of 50 female fans said that they both favor the new team mascot. If 1 male and 1 female fan are randomly selected, what is the probability that both favor the new mascot? P(M and F) = (50/75) * (40/50) = 8/15 b. For a fundraiser, a class sells 150 raffle tickets for a mall gift certificate and 200 raffle tickets for a booklet of movie passes. You buy 5 raffle tickets for each prize. What is the probability that you win the mall gift certificate but NOT the booklet of movie passes? P(WM and Lm) = (5/150) * (195/200) = 13/400 c. A survey found that 46% of parents surveyed say that they read to their children at least once a week. If 3 parents are selected at random, what is the probability that all 3 will say that they read to their children at least once a week? P(All 3) = 0.46 * 0.46 * 0.46 = 0.097 = 9.7% d. A spinner is divided into ten equal regions numbered 1 to 10. What is the probability that 3 consecutive spins result in perfect squares? P(All 3) = (3/10) * (3/10) * (3/10) = 27/1000 e. What is the probability that at least one of the spins is a perfect square? P(At least 1) = 1 - P(None) = 1 - (7/10) * (7/10) * (7/10) = 657/1000 This worksheet focuses on probability, specifically using complements to calculate probabilities of events. It covers various scenarios, including dice rolls, birthdays, and real-life events like surveys and raffles.

Math Algebra Probability Notes 5 Key

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