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Probability

Probability. Basic Concepts. Experiment . An activity or measurement that results in an outcome we cannot predict with absolute certainty. A coin is flipped twice. We conduct a 10 year medical study on 12,000 participants. Event. The simplest outcome of an experiment Toss two coins:

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Probability

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  1. Probability Basic Concepts

  2. Experiment • An activity or measurement that results in an outcome we cannot predict with absolute certainty. • A coin is flipped twice. • We conduct a 10 year medical study on 12,000 participants

  3. Event • The simplest outcome of an experiment • Toss two coins: • 4 possible outcomes for each toss • Each possible outcome is an event • All the possible outcomes comprise the…

  4. Sample space • All the possible outcomes of an experiment: • (HH) • (HT) • (TH) • (TT) • More examples • One coin • One die • Red, white, and blue socks • With and without replacement

  5. Probability measure • A function, P, from the subsets of the sample space (Ω) that satisfies the following axioms: • P(Ω) = 1; that is, the sum of all the probabilities in the sample space will = 1 • P(A)≥0 for any event, A, that is included in the sample space (Ω). • For mutually exclusive events A1 and A2 within the same sample space P(A1UA2) = P(A1) + P(A2)

  6. Classical Probability • Proportion of times an event can be expected to occur • All outcomes are equally likely:

  7. Examples • Flip a coin twice: • P(HH) = P(HT) = P(TH) = P(TT) = 0.25 • Roll a die: • P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

  8. Examples • Consider events A and B (coins) • We can determine their probabilities by adding the probability of each occurrence. • A: exactly one head [(HT), (TH)] • P(A) = 0.25 + 0.25 = 0.50 • B: first flip head [(HH),(HT)] • P(B) = ¼ + ¼ = ½ • If we have an experiment where we roll one die, what is the probability of rolling less than a 3? • Sample space • Event • P(A)

  9. Compound Events • Mutually Exclusive Events • If one event occurs, the other cannot • Exhaustive Events • Set of events that include all the possible outcomes of an experiment • Events are exhaustive because one of them must occur • When events are mutually exclusive and exhaustive, sum of their probabilities must equal 1

  10. Intersection and Union • Intersection of Events • Two or more events occur at the same time in the same experiment. [A and B, or A and B and C] • Union of Events • At least one of a number of possible events occurs in the same experiment. [A or B, or A or B or C]

  11. What’s the probability? Experiment: the throw of one die A: (observe an even number) B: (observe a number <=3) • Describe A union B • Describe A intersection B • Calculate the probabilities of 1. and 2.

  12. Example: Unions and Intersections

  13. Addition Rules • When events are mutually exclusive • When events are not mutually exclusive

  14. Practice Identity 2 events that are mutually exclusive Identify 2 events that intersect

  15. Practice What is the probability that a household would be in the South or Midwest or have internet access? What is the probability a household would be in the West and not have Internet access?

  16. Venn Diagram A = 0.45 0.05 B = 0.15 What is the probability of Ac? What is the probability of Bc? What is the probability of AUB?

  17. Probabilities • Marginal Probability • Probability a given event will occur. No other events are considered. P(A) • Joint Probability • Probability that two or more events will all occur. P(A and B) • Conditional Probability • Probability that an event will occur given that another event has already occurred. P(A|B)

  18. Multiplication Rules • Independent Events • The occurrence of one has no effect on the probability that the other will occur. • Dependent Events • The occurrence of one event influences the probability of the other.

  19. Multiplication Rules • When events are independent • When events are not independent

  20. Fireworks Chart

  21. Conditional Probability

  22. Conditional Probability Problem • A corporation is going to select 2 of its regional managers for promotion to VP. They have 6 male and 4 female regional managers. Assume each manager has an equal probability (1/10) of being selected. • What is the probability that both people selected for regional manager are male?

  23. Practice Problems • A fair coin is tossed 4 times. What is the probability of getting at least one tail? • What is the probability of getting exactly one head? • A card is drawn for a standard deck. What is the probability that card will be a jack or a king?

  24. More Practice Problems • A standard pair of 6-sided dice is rolled. What is the probability of rolling a sum greater than or equal to 3? • Three cards are drawn with replacement from a standard deck. What is the probability that the 1st card will be a diamond, the 2nd card will be black, and the third card will be a queen?

  25. Still more practice problems • 2 cards are drawn without replacement from a standard deck. What is the probability of choosing a club and then a black card? • A box contains 6 green marbles and 19 white marbles. If the first marble chosen is white, what is the probability of choosing a white marble on the second draw?

  26. Counting • Principle of multiplication • m ways for event 1 to happen • n ways for event 2 to happen • Total number of possibilities = m x n • If each of k independent events can happen n different ways, the total number of possibilities is nk

  27. Counting • Factorial rule of counting • n! = n x (n-1) x (n-2) x … x 1 • 0! = 1

  28. Counting • Permutations: Number of possible arrangements of n items in order

  29. Counting • Combinations: Order doesn’t matter. We consider only the possible set of objects.

  30. Counting Problems • A 29-sided die is rolled 2 times. How many different outcomes are possible? • License plates consist of 2 letters followed by three numbers. Duplicate digits are allowed. How many different outcomes are possible? • A doctor visits her patients during morning rounds. In how many ways can she visit the 8 patients?

  31. More Counting Problems • A coordinator will select 6 songs from a list of 8 songs to compose a musical lineup. How many different lineups are possible? • 4 cards are chosen from a standard deck. How many different 4 card hands are possible? • A person tosses a coin 16 times. In how many ways can he get 6 heads?

  32. Practice • The daily number in a state lottery is a 3-digit integer between 000 and 999. • What is the probability that the winning number will be 555? • Today’s winning number is 347. You are going to buy a ticket tomorrow and you plan to select number 347. Is this a good idea? Why or why not?

  33. Practice Your company has two computer systems available for processing telephone orders. Computer system A has a 10% chance of being down; computer system B has a 5% chance of being down. The computer systems operate independently. At least one system needs to work in order to process new orders. For a typical telephone order, determine the probability that: • Neither computer system will be operational. • Both computer systems will be operational. • Exactly one of the computer systems will be operational. • What is the probability that the order can be processed without delay?

  34. Practice • A security service employing 10 officers has been asked to provide 3 officers for crowd control at a local carnival. In how many different ways can the firm staff this event?

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