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The Laws of Probability

The Laws of Probability. Basic Laws to review. Probability of an event : Complement of an event: P(A’) = 1 – P(A) The complement of event A is that the event A does not occur.

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The Laws of Probability

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  1. The Laws of Probability

  2. Basic Laws to review • Probability of an event : • Complement of an event: P(A’) = 1 – P(A) The complement of event A is that the event A does not occur.

  3. If the probability of rolling a ‘2’ on a six sided die is one sixth, what’s the probability of not rolling a 2? (Enter a fraction) • 5/6 • 0.0

  4. B A Disjoint Events • Two events A and B are disjoint or mutually exclusive if they have no outcomes in common and can never happen simultaneously. Using a Venn Diagram disjoint or mutually exclusive events are shown: In Algebra: P(A or B)=P(A U B) = P(A) + P(B) P(A and B) = P(A ∩ B) = 0

  5. Example: • A cooler contains 20 bottles made up of 8 cokes, 5 pepsi’s, and 7 waters. The probability of choosing a coke or pepsi is:

  6. You Do: For the junior class picnic, parents prepare hotdogs, hamburgers, or bar – b – que sandwiches. They have 100 hot dogs, 55 hamburgers, and 90 Bar – b – que sandwiches. What is the probability that you will get a hamburger or a bar – b – que sandwich? (Click in your answer as a reduced fraction) • 29/49 • 0.1

  7. General Addition Rule: • If two (or more) events are not disjoint the above formula doesn’t work. The rule is: P(A or B) = P(A) + P(B) – P(A and B) • EXAMPLE: How many people have a dog? How many people have a cat? How many people have a dog and a cat? D C

  8. Example • Example: In a class there are 12 boys made up of 8 senior and 4 juniors. There are also 8 girls, made up of 3 seniors and 5 juniors. Find the probability of choosing a boy or a senior. • Note – choosing a boy and choosing a senior are not disjoint (they can occur simultaneously). So the probability of choosing a boy or a senior = P(boy) + P(senior) – P(senior boy).

  9. You Do: • In a class there are 12 boys made up of 8 senior and 4 juniors. There are also 8 girls, made up of 3 seniors and 5 juniors. Find the probability of choosing a girl or a junior. (Click in a reduced fraction.) • 3/5 • 0.0

  10. A rental car lot has 49 American made cars and 26 Foreign cars. Of the American cars, 35 of them are white and of the foreign cars, 15 are white. A car is chosen at random. Find the probability that the car is American or White. • 64/75 • 0.0

  11. 8/15 • 0.0 A rental car lot has 49 American made cars and 26 Foreign cars. Of the American cars, 35 of them are white and of the foreign cars, 15 are white. A car is chosen at random. Find the probability that the car is Foreign or not White. (May need a chart)

  12. A pizza shop has two sizes of pizzas, large and small. On a certain day, a pizza shop made 59 plain pizza and 72 pizzas with toppings. Of the 59 plain pizzas, 19 were small and of the 72 pizzas with toppings, 42 were large. A pizza is chosen at random. Find the probability that the pizza is small or plain. (Make a chart) • 89/131 • 0.0

  13. A pizza shop has two sizes of pizzas, large and small. On a certain day, a pizza shop made 59 plain pizza and 72 pizzas with toppings. Of the 59 plain pizzas, 19 were small and of the 72 pizzas with toppings, 42 were large. A pizza is chosen at random. Find the probability that the pizza is large or with toppings. (Make a chart) • 112/131 • 0.0

  14. Independent Events • Two events are independent if knowing one occurs doesn’t change the probability of the other occurring. This is the non – mathematical definition. • EXAMPLE: Tossing one coin and then another coin are independent events. The result of the second coin toss has nothing to do with the results of the first coin toss.

  15. Proving Independence • Proving independence is difficult. For instance, is making a second foul shot in basketball independent of the first foul shot? What do you think? • Mathematical proof of independence: If two events are independent then: • P(A) P(B) = P(A and B) • ALSO, IF P(A) P(B) = P(A and B), THEN the two events are independent.

  16. Example • Using the previous example, are choosing a senior and choosing a boy independent? • If so, P(senior) X P(Boy) = P(Senior boy) • Fill in the probabilities on the left side of equation; • Fill in the probabilities on the right side of equation; • Are the two sides equal? • What does this tell you?

  17. At Parkland High School, there are 13 math teachers. There are 5 men and 9 women. Of the 5 men, 2 have their national board certification and of the 9 women 1 has her national board certification. Is choosing a teacher with national board certification and a woman independent? • Yes • No

  18. Conditional Probability • Definition: The probability of one event happening, given that another event has happened. • Notation: P(B|A) means the probability of B occurring given that A occurred.

  19. Example Find P(Boy|Senior) – the probability of choosing a Boy given that we chose a senior. There are 11 seniors and 8 of them are boys, so P(B|S) = Find P(Senior|Boy) – the probability of choosing a senior, given that we chose a boy. There are 12 boys, and of those 12, 8 are seniors, so P(S|B) =

  20. THE FORMULA

  21. YOU DO: • 5/8 • 0.0 Find the P(Junior|Girl)

  22. You Do: • 1/3 • 0.0 • Find the P(Junior|Boy)

  23. You Do: • 3/11 • 0.0 Find the P(Girl|Senior)

  24. Example: • Let a pair of fair dice be tossed. Make a sample space of each possible toss.

  25. Example cont… • Find the probability that at least one of the die is a four. • Find the probability that the sum is a 7 • Find P(the one of the die is a 4|sum is 7) • Given that at least one of the dice is 4, find the probability that the sum is 7. • Are rolling two dice with at least one of them a 4 and the sum being 7 A) mutually exclusive or B) independent. Why?

  26. A cooler has 12 Coke’s and 15 Pepsi’s. 9 of the Cokes are diet Coke’s and 5 of the Pepsi’s are diet Pepsi’s. Make a chart of what is in the cooler and find the probability that the bottle is a Coke, given that the bottle is a diet drink. • 9/14 • 0.1

  27. A cooler has 12 Coke’s and 15 Pepsi’s. 9 of the Cokes are diet Coke’s and 5 of the Pepsi’s are diet Pepsi’s. Given that the bottle is a Pepsi, find the probability that the bottle is a Diet Pepsi • 1/3 • 0.0

  28. Are choosing a regular (non diet) and choosing a diet drink independent or mutually exclusive? Why? • Independent • Mutually exclusive [Default] [MC Any] [MC All]

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