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Probability

Probability. Questions, comments, concerns? Ok to move on?. Vocab. Trial- number of times an experiment is repeated Outcomes- different results possible Frequency- number of times an event occurs Relative Frequency- ??. Warm Up. List all the possible outcomes for tossing a coin once….

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Probability

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  1. Probability

  2. Questions, comments, concerns? Ok to move on?

  3. Vocab • Trial- number of times an experiment is repeated • Outcomes- different results possible • Frequency- number of times an event occurs • Relative Frequency- ??

  4. Warm Up List all the possible outcomes for tossing a coin once… Head, Tail or HT Let an event be tossing a single coin two times. Represent the sample space : a. In a list form b. in a table form c. as a tree diagram. HH, HT, TH, TT

  5. Practice An urn contains 5 purple, 3 white, and 4 red marbles. Two marbles are drawn. Draw a sample space for the event using a TREE DIAGRAM.

  6. Practice • Let the event be a couple having children. Draw a tree diagram for the outcomes of a couple having four children.

  7. Practice Let the event be tossing a coin and then rolling a fair six-sided die. Draw a tree diagram and list the outcomes

  8. Practice • The town of Alright has a population of 200 adults: 60% are women and 10% of them are more than 6ft tall. Of the men, 30% are more than 6 ft tall. Draw and label a tree diagram of the town.

  9. NOW the juicy stuff… Probability! Probability: Formula in set notation? Theoretical vs Experimental Probability?

  10. Practice Find the probability that a 7 will show when rolling a fair six-sided die whose faces are numbered 1 through 6.

  11. Practice • Find the probability that an even number shows when a six-sided die is rolled.

  12. Review A compliment is…. P(A’) = 1- P(A) Ex: If there is a 35% chance it will rain tomorrow, what is the probability it will NOT rain.

  13. Practice Find the probability that when rolling two dice they will not show doubles.

  14. Laws of Probability Mutually Exclusive: P (A U B) = P(A) + P(B) Combined Events: P(A U B) = P(A) + P(B) – P(A∩ B) Independent Event: P(A ∩ B) = P(A) ∙ P(B) Dependent Event: P (B|A) = Note: OR = Union AND = Intersection

  15. Mutually Exclusive If it is not possible for the events to occur at the same time.

  16. Mutually Exclusive Probability • P(A U B) = P(A) + P(B) • where sets A and B have no elements in common.

  17. Example Let A be the event that a person is a female and B be the event that a person is 1.6 metres tall. • Are the two mutually exclusive? • Justify your answer by drawing a venn diagram

  18. Food for thought Can two political parties be considered mutually exclusive?

  19. Practice Find the probability of having a 5 or an even number show when rolling a six-sided die.

  20. Combined Events • These events are NOT mutually exclusive.

  21. Combined Events Probability P(A U B) = P(A) + P(B) – P(A ∩ B) Consider the following: U= numbers 1-12. A = multiples of 3 numbers. B= Even numbers. Create a Venn diagram. What is P(A U B)?

  22. Practice • Find the probability, when drawing a single card from a standard 52-card deck, that a card is either a nine or a face card.

  23. Practice • Find the probability, when drawing a single card from a 52-card deck, that a card is either red or an ace.

  24. Practice • If P(A) = .37, P(B) = .55, P(A∩B) = .13, find: • P(AUB) • P(A’) • P(B’) • P(A∩B’)

  25. Data was collected for which gender reads which types of book. A book as selected at random. Find the probability that: a. The book was read by a female or it was an action book. b. The book was a historical book or it was read by a male. c. The book was not read by a male and it was an action book.

  26. Independent Events This occurs if the probability of the first event does NOT effect the probability of the second event… Then to see what the probability of both happening is P(A∩B) = P (A) ∙ P(B)

  27. Practice If the probability it will snow on Monday is 25% and on Tuesday it is 40%, what is the probability that it will snow on both Monday and Tuesday?

  28. Practice Find the probability of the toss of a coin landing heads up and the roll of a die showing a four.

  29. Practice • An urn contains 4 red marbles and 3 green marbles. Find the probability of drawing a red marble, replacing it, drawing a green marble, replacing it, and then drawing a red marble.

  30. From a group of 20 athletes it is found that 13 played billiards, 12 played golf and 5 played both billiards and golf. • Draw a venn diagram to represent this information. • Find the probability that an athlete chosen at random • Plays golf • Does not play billiards • Plays billiards and golf • Plays billiards or golf • State a reason as to whether or not the events are independent. • State a reason as to whether or not the events are mutually exclusive.

  31. Dependent Events • Two events are said to be dependent if the probability of the first event occurring influences the probability of the second event occuring.

  32. Example An urn contains 4 red marbles and 3 green marbles. Find the probability of drawing a red marble, NOT replacing it, and then drawing a green marble.

  33. Practice When drawing two cards from a standard 52-card deck, find the probability of both cards being black.

  34. A bag contains two red sweets and three green sweets. Jacques takes one sweet from the bag, notes its colour, then eats it. He then takes another sweet from the bag. Copy and complete the tree diagram below to show all possibilities.

  35. Conditional Probability Recognizable by the words “given that” P(B|A) =

  36. Practice • Six identical marbles, numbered, 1,2,3,4,5,6 are placed in an urn. A single draw is made. It is known only that an even-numbered marble is selected. What is the probability that it is a four?

  37. Practice Consider the data collected: • Find the probability that the kind of vehicle driven to school was a truck, given that the student was a senior. • Given that the vehicle driven to school was a sports car, find the probability that the student was a junior.

  38. Practice From a standard 52 card deck, let event D be the selection of a diamond and event F be the selection of a face card. One card is selected at random. Find the probability that: a. P(the card is a diamond or a face card) b. P(the card is neither a diamond nor a face card) c. P(a face card is selected given that a diamond has already been selected.) d. P(a diamond is selected given that a face card has already been selected. )

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