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PROBABILITY

PROBABILITY. Section 3.1: Basic Concepts of Probability and Counting. Probability Experiment (or experiment) – an action or trial through which specific results are obtained Outcome – result of a SINGLE trial in an experiment Sample Space – list of ALL possible outcomes

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PROBABILITY

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  1. PROBABILITY Section 3.1: Basic Concepts of Probability and Counting

  2. Probability Experiment (or experiment) – an action or trial through which specific results are obtained Outcome – result of a SINGLE trial in an experiment Sample Space – list of ALL possible outcomes Event – Any subset of the sample space

  3. Find the sample space for when you flip a coin AND roll a die. - Find the sample space using a TREE DIAGRAM

  4. Simple Event – an event that consists of only ONE outcome Examples: Flip a heads Roll a 3 Spin a 10 NON-Examples: Roll an even number Pick a red card

  5. Find the Sample Space: • 6 outcomes to rolling a die • 8 outcomes to spinning a spinner • How many outcomes are there to rolling the die AND spinning the spinner?

  6. FUNDAMENTAL COUNTING PRINCIPLE If there are m ways of doing one thing and n ways of doing another, then there are m * n ways of doing both. Easy way to find size (number of items) of a sample space.

  7. Example: The access code to the laptop cart is four digits between 0 and 9. How many possible combinations are there? Use the blank space method. What if the code begins with an even number?

  8. Example: License plates are made from 7 characters: either a letter or a number. How many license plates can exist if: The first 3 must be letters and the next 4 numbers? The first 3 must be letters and the next 4 numbers, NONE OF WHICH MAY BE REPEATED? The first 3 must be letters and the next 4 numbers, where the first letter CANNOT be A, E, I, O, or U AND no numbers can be EVEN?

  9. Classwork/Homework Complete The Fundamental Counting Principle Worksheet

  10. Notation: P(E) Probability of the event E happening P(E) = number of ways E can happen number of possible outcomes

  11. FACTS ABOUTS PROBABILITY Probabilities are between 0 and 1 inclusive P(E) = 0 mean E is impossible P(E) = .5 means there is an even chance of E happening as not happening P(E) = 1 means event E is CERTAIN to happen

  12. Complement of an Event, E • Set of all outcomes in the sample space that are NOT in E • Notation: P(E’) or P(not E) • P(E’) = 1 – P(E)

  13. Complement of an Event, E • Event: rolling a number that is AT LEAST 5 • Complement of Even: rolling a number less than 5 • P(E) = • P(E’) = • P(E) + P(E’) = 1

  14. If we have a lottery with 1000 tickets and you buy 1 ticket. What is the probability that you win? What is the probability that you don’t?

  15. Probability Examples In a deck of 52 cards, what is the probability of drawing a red card? If we roll a 6 sided die, what is the probability of rolling less than a 3? If we roll two 6 sided dice, what is the probability of rolling a sum of 7? If we roll two 6 sided dice, what is the probability of rolling a sum less than a 4?

  16. Guided Practice: Probability and Counting Worksheet

  17. Class Activity/HW: Pg. 142 #5 - #16, #21 - #24, #27 - #35

  18. Stations of Probability Activity • 2 experiments run in each group • 10 minutes/experiment • Experimenter/recorder (switch halfway through each experiment) • 3 minutes to answer questions • STAY ON FRONT OF WORKSHEET • EMPIRICAL – probability from EXPERIMENT • THEORETICAL – in “theory” what probability should be for the given event

  19. COMBINED SIDE OF WORKSHEET • With group across from you, combine data for BOTH experiments • Answer the questions. • FOCUS: Compare results to questions on both sides. What seems to be happening? • Bring observations to class tomorrow.

  20. 2 Types of Probability Theoretical – number of desired outcomes number of total outcomes 2. Empirical – number of times event E happens number of attempts

  21. Observations from yesterday’s activity:

  22. The Law of Large Numbers: As an experiment is repeated over and over enough times, the empirical probability of an event approaches the theoretical probability.

  23. Section 3.2 Conditional Probability Multiplication Rule

  24. Conditional Probability • Used to find the probability of sequential events • THE PROBABILITY OF AN EVENT OCCURING, GIVEN THAT ANOTHER EVENT HAS ALREADY OCCURRED.

  25. Notation: P(B|A) PROBABILITY OF B, GIVEN A The conditional probability of event B given that event A has already occurred.

  26. Two cards are selected from a deck of cards, one after the other, WITHOUT REPLACEMENT. What is the probability that: The second card drawn is a King given that the first card drawn was a Queen? The second card drawn is a King given that the first card drawn was a King?

  27. Find the probability that a child has a high IQ. Find the probability that a child has a high IQ given that the child has gene X. Find the probability that a child has NO gene X. Find the probability that a child has a NO gene X given that the child has a normal IQ.

  28. Independent Events • Event taking place that have NO effect on the probability of the other one occurring. • They do NOT “INTERFERE” with each other. • For Independent events, • P(B|A) = P(B) • P(A|B) = P(A) • Events that are NOT independent are dependent.

  29. Independent or Dependent? Tossing a coin and rolling a 2. Selecting an Ace replacing it into the deck and then selecting a King. Selecting an Ace WITHOUT REPLACEMENT and then selecting a King.

  30. The Multiplication Rule The probability of two events A and B happening in sequence is P(A and B) = P(A) * P(B|A) Thus, for independent events, P(A and B) = P(A) * P(B).

  31. P(roll a 5 and a 6) P(King and 10) WITH REPLACEMENT P(King and 10) WITHOUT REPLACEMENT P(red and black) WITHOUT REPLACEMENT

  32. P(successful surgery) = .75 Surgeries are independent. There are four surgeries today. Find: P(4 successful surgeries) P(0 successful surgeries) P(at least 1 successful surgery)

  33. 65% of jury is female. 1 out of 4 women works in the health field. Find: P(Female and in health field) P(Female and no in health field)

  34. Classwork/Homework Pg. 154 – 155 #5 - #8, #15, #16, #25, #26, #27

  35. Section 3.3 The Addition Rule

  36. Section 3.2: P(A|B) P(A AND B ) Section 3.3: P(A OR B)

  37. Mutually Exclusive Events – Two events A and B are MUTUALLY EXCLUSIVE if A and B CANNOT occur at the same time. Example: Venn Diagrams

  38. Examples: Decide if the events are mutally exclusive. Event A: Roll a 3 on a die. Event B: Roll a 4 on a die. 2. Event A: Randomly select a male student. Event B: Randomly select a nursing major. 3. Event A: Randomly select a blood donor with type O blood. Event B: Randomly select a female blood donor.

  39. The Addition Rule: The probability that events A OR B will occur, P(A or B) is given by P(A or B) = P(A) + P(B) – P(A and B) If events A and B are mutually exclusive then, P(A or B) = P(A) + P(B).

  40. Examples: You select a card from a standard deck. Find the probability that the card is a 4 OR an ace. You roll a die. Find the probability of rolling a number less than three or rolling an odd number. You roll a die. Find the probability of rolling a 6 or an odd number. You select a card from a standard deck. Find the probability that the card is a face card or a heart.

  41. Blood Type RH-Factor Find the probability that the donor has type O blood or type A blood. Find the probability that the donor has type B blood or is RH-negative. Find the probability that the donor has type B blood or type AB blood. Find the probability that the door has type O blood or is RH-positive.

  42. Pg. 164 Summary chart of ALL probability equations and definitions. DO: 1. Read through Example 5. Understand. 2. Do Try it Yourself #5.

  43. RH Factor/Blood Type Worksheet Complete FRONT side #1 - #5 KEEP WORKSHEET!

  44. Classwork/Homework Assignment: Due FRIDAY Pg. 165 – 169 #1 - #12 #14, #17, #18 #21 - #24

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