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Finding Probabilities

Finding Probabilities. Sec 5.2 Part one. Vocabulary: . Sample Space : For a random Phenomenon the sample space is the set of all possible outcomes. Examples: For the flip of two coins the sample space is. {(HH), (HT), (TH), (TT)} A single role of a six sided die.

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Finding Probabilities

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  1. Finding Probabilities Sec 5.2 Part one

  2. Vocabulary: • Sample Space: • For a random Phenomenon the sample space is the set of all possible outcomes. • Examples: For the flip of two coins the sample space is. {(HH), (HT), (TH), (TT)} A single role of a six sided die. {1, 2, 3, 4, 5, 6} Types of pets people have. {Dogs, Cats, Birds, Fish, reptiles, Insects, mollusks}

  3. Vocabulary: • Event: • An event is a subset of the sample space. • Example: For the flip of two coins an event could be two heads. A single role of a six sided die rolling a 5 would be an event. For pets the selection of a mammal could be an event.

  4. Example Scenario: • You are building an experiment testing the effects of caffeine, sugar, and alcohol on memory. You are going to randomly assign each of your subjects to take one or all of these substances by flipping a coin for each. • Lets look at the sample space for our subjects.

  5. Sample Space: Sample Space CSA CSN CNA CNN NSA NSN NNA NNN Tree Diagram

  6. Events: • So with our sample space; {CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN} There are many different types of possible events. Examples: 1) CNA 2) Takes caffeine 3) Does not take caffeine 4) Takes either sugar or caffeine but not alcohol 5) Takes nothing (Control)

  7. So, let find some probabilities: • First, some things that must be true about our probabilities. • The probability of each individual outcome is between zero and one. • The total of all the individual probabilities equals ONE.

  8. Probability of an event: • The probability of an event A, denoted P(A), is the ratio between the number of outcomes in that event and the number of outcomes within the sample space. When all possible outcomes are equally likely!!!! • The probability of an event B, denoted P(B), is found by adding the probabilities of the individual outcomes.

  9. So back to our sample space; {CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN} Question: What is the probability a subject got caffeine or sugar? To Answer: How Many outcomes had either caffeine or sugar? 6 How many outcomes are in the sample space? 8 So, the probability of this event is; So, the probability that a given subject was given either caffeine or sugar is .75 (or 75%). Either notation is ok with me.

  10. Again given our sample space; {CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN} Question: What is the probability a subject got caffeine or sugar not both? To Answer: How Many outcomes had either caffeine or sugar not both? 4 How many outcomes are in the sample space? 8 So, the probability of this event is; So, the probability that a given subject was given either caffeine or sugar is .5 (or 50%).

  11. Yet again given our sample space; {CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN} Question: What is the probability a subject receiving only two compounds? To Answer: How Many outcomes have only two compounds? 3 How many outcomes are in the sample space? 8 So, the probability of this event is; So, the probability that a given subject was given either caffeine or sugar is .375 (or 37.5%).

  12. Probabilities from trials: (remember contingency tables) • A sample of 3000 registered voters is taken to find the probability of certain events by party affiliation. The data is summarized below.

  13. Questions: • What is the sample space. {(R,V), (R,NV), (D,V), (D,NV), (O,V), (O,NV)} • What is the probability that a selected subject voted? 1426/3000 • What is the probability that a selected subject was a Democrat? 1454/3000 More Questions??

  14. Your Questions so far…

  15. Dealing with Paired events 5.2 Part two.

  16. Vocabulary: • Complement of an event; • The complement of an event A is made up of all outcomes in the sample space that are not within A. Denoted ~A or . • The sum of the probabilities of an event and its complement always add to one. • Thus, P(~A)=1- P(A)

  17. Examples of Complements • Let A be the event of rolling a 6 on a fair die. • Then ~A would be… • Let A be the event of having at least one woman on the supreme court. • Then ~A would be… • The event of being in the bubble • Then ~A would be...

  18. Venn Diagrams: Visual representations of sets; U U A B The Sample Space, “Universe” Bubbles within are events. “Sets”

  19. Venn Diagrams: Complements U A U ~A An event A. The event ~A

  20. Venn Diagrams: Set interaction; U A B U A B Disjoint events, Non-disjoint events.

  21. Disjoint Events: Recall our scenario with the three compounds. Goal: Make a Venn diagram with event A as getting one compound and event B getting two compounds. CSN NNA CSA CNA NSN NNN NSA CNN A B Are these disjoint events? What is ~A? What is ~B?

  22. Operations on Events: Union and intersections; Given events A and B we can find the... U A B U A B Intersection of A and B, Union of A and B.

  23. Intersections: “AND” • The intersection of A and B is made up from events in both A and B. • Is this more or less of a restriction? • Is this going to make our Probabilities bigger or smaller? • Link “and” with shrinking probabilities

  24. Unions: “OR” • The union of A and B is made up from events in either A or B. • Is this more or less of a restriction? • Is this going to make our Probabilities bigger or smaller? • Link “or” with growing probabilities.

  25. Operations on Events: Union and intersections; Given events A and B we can find the... U A B U A B Intersection of A and B, “A and B” Union of A and B. “A or B”

  26. Probabilities for unions: • To find the probability of Event A or Event B occurring we must use unions. Case 1: If the events are disjoint find the sum of the probabilities of events A and B. P(A or B) = P(A) + P(B) Case 2: If the events are not disjoint add the probabilities of the two events and subtract the probability of their intersection. P(A or B) = P(A) + P(B) – P(A and B)

  27. Why do we subtract the intersection? U B A Take Notes on board work for complements, unions.

  28. Probabilities for intersections: • For independent events; • To find the probability of Event A and Event B occurring we must use intersections. • The probability of being within an intersection is the product of the event probabilities. P(A and B) = P(A) * P(B)

  29. Examples: U B A Take Notes on board work intersections and more.

  30. Recall our Experiment: In our experiment we made each event equally likely. What if we wanted the change this. In order to make our control group larger lets role a die instead of flipping a coin. For each compound if the die comes up four or less they get a dose. If the die is five or six they don’t get that compound. What is the Probability of getting all three compounds? What is the probability of getting caffeine or Alcohol? What is the Probability of getting at least two compounds? Let Map this out on our Hand outs.

  31. Summary: • Probabilities are between zero and one. • The sum of the all probabilities is one. • P(A) + P(~A) = 1 • The union: P(A or B) = P(A) + P(B) – P(A or B) • For Independent the intersection: P(A and B) = P(A)*P(B) • Events are disjoint if they have no common element.

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