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Chapter 4

Chapter 4. Probability and Counting Rules. Section 4-1. Sample Spaces and Probability. Learning Target. Determine sample spaces and find the probability of an event using classical probability or empirical probability. Basic Concepts.

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Chapter 4

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  1. Chapter 4 Probability and Counting Rules

  2. Section 4-1 Sample Spaces and Probability

  3. Learning Target • Determine sample spaces and find the probability of an event using classical probability or empirical probability.

  4. Basic Concepts • Probability Experiment – a chance process that leads to well-defined results called outcomes • Outcome – the result of a single trial of a probability experiment • Sample Space – the set of all possible outcomes of a probability experiment • Examples: tossing a coin (head, tail), roll a die (1,2,3,4,5,6), Answer a true/false question (true, false)

  5. Sample Space for Rolling Two Dice

  6. Cards in a Regular Deck of Cards • 4 suits – spades, diamonds, hearts, clubs • 13 of each suit • There are 3 face cards in each suit – jack queen, king • 52 cards total

  7. Gender of Children • Find the sample space for the gender of the children if a family has three children. • How can this be done?

  8. Tree diagram • Device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment.

  9. Outcomes BBB BBG BGB B BGG 1st child 2nd child 3rd child GBB G GBG GGB GGG

  10. More Vocab • Event – consists of a set of outcomes of a probability experiment • Simple Event – an event with one outcome • Compound Event – event with more than one outcome • Example: the event of rolling an odd number on a die

  11. Classical Probability • Uses sample spaces to determine the numerical probability that an event will happen • Assumes that all outcomes are equally likely to occur • Formula – number of outcomes in E divided by the total number of outcomes in the sample space,

  12. Probabilities can be expressed as fractions, decimals, or percents. Most problems will be expressed as fractions or decimals. If the problems starts in fractions the answer should be a fraction. If the problem starts as a decimal the answer should be a decimal. • Fractions should always be reduced and decimals rounded to two or three decimal places.

  13. Practice Problems • A card is drawn at random from an ordinary deck of cards. Find these probabilities. • Of getting a jack • Of getting a red ace • Of getting the 6 of clubs • Of getting a 3 or a diamond • Of getting a 3 or a 6 • If a family has three children, what is the probability that two of the three children are girls?

  14. Solutions

  15. 4 Probability Rules • The probability of any event E is a number (either a fraction or decimal) between and including 0 and 1. This is denoted by • If an event E cannot occur (i.e., the event contains no members in the sample space), its probability is 0. • If an event E is certain, its probability is 1. • The sum of the probabilities of all the outcomes in the sample space is 1.

  16. Complementary Events • The complement of an event E is the set of all outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by E (read “E bar”) • Example: The event E of getting an odd number is 1,3,5. The complement of E is getting an even number (2,4,6).

  17. Practice Problems • Find the complement of each event. • Rolling a die and getting a 4 • Selecting a letter of the alphabet and getting a vowel • Selecting a month and getting a month that begins with a J • Selecting a day of the week and getting a weekday

  18. Solutions • Getting a 1,2,3,5,6 • Getting a consonant (assume y is a consonant) • Getting February, March, April, May, August, September, October, November, or December • Getting Saturday or Sunday

  19. Rule for Complementary Events • or or

  20. Empirical Probability • Relies on actual experience to determine the likelihood of outcomes • Given a frequency distribution, the probability of an event being in a given class is

  21. Practice Problems • In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. • A person has type O blood • A person has type A or type B blood • A person has neither type A nor type O blood • A person does not have type AB blood

  22. Solution A. B. C. D.

  23. Applying the Concepts and Exercise 4-1

  24. Section 4-2 The Addition Rules for Probability

  25. Learning Target • IWBAT find the probability of compound events, using the addition rules.

  26. Mutually Exclusive • Two events are mutually exclusive if they cannot happen at the same time. • In other words they have no outcomes in common. • Example: getting a 4 and a 6 are mutually exclusive.

  27. Which ones are mutually exclusive? • Getting an odd number and getting an even number • Getting a 3 and getting an odd number • Getting a 7 and a jack • Getting a club and getting a king

  28. Addition Rule #1 • When two events are mutually exclusive, the probability that A or B will occur is

  29. Practice Problems • A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a doughnut at random, find the probability that either is a glazed or chocolate doughnut. • At a political rally, there are 20 republicans, 13 democrats, and 6 independents. If a person is selected at random, find the probability that he or she is either a democrat or an independent.

  30. Answers

  31. Addition Rule #2 • If A and B are not mutually exclusive, then

  32. Practice Problems • In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

  33. Answer

  34. For 3 events • Mutually Exclusive • Not Mutually Exclusive

  35. Venn Diagrams P(A) P(B) P(A and B) Mutually Exclusive P(A) P(B) Not Mutually Exclusive

  36. Exercises 4-2 1-25 odd and #8

  37. Graded for correct answer • 2, 6, 10, 14, 18, 20, 24, 26

  38. Section 4-3 The Multiplication Rules and Conditional Probability

  39. Learning Target • IWBAT find the probability of compound events, using the multiplication rule.

  40. Multiplication Rules • The multiplication rules are used to find the probability of events that happen in sequence. • For example, when you toss a coin and roll a die, you can find the probability of flipping a head and rolling a 4. • The events are independent since the outcome of the first event does not effect the second.

  41. Rule #1 • When two events are independent, the probability of both occurring is

  42. Examples • A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. • A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen then an ace.

  43. More Examples • An urn contains 3 red marbles, 2 blue marbles, and 5 white marbles. A marble is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these. • Selecting 2 blue marbles • Selecting 1 blue marble then 1 white marble • Selecting 1 red marble then 1 white marble

  44. Solutions

  45. The multiplication rule can be extended to three or more events by using the formula…

  46. Dependent Events When the outcome or occurrence of the first event effects the outcome or the occurrence of the second event in such a way that a probability is changed, the events are said to be dependent. When situations involve not replacing the item that was selected first, the events are dependent.

  47. Conditional Probability • The conditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred.

  48. Rule #2 • When two events are dependent, the probability of both occurring is Means that B happens given that A happened first. (Conditional Probability)

  49. Examples • Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events. • Getting 3 jacks • Getting an ace, a king, and a queen in order • Getting a club, a spade, and a heart in order • Getting 3 clubs

  50. Solutions

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