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PROGRAMME 28

PROGRAMME 28. PROBABILITY. Programme 28: Probability. PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events

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PROGRAMME 28

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  1. PROGRAMME 28 PROBABILITY

  2. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  3. Programme 28: Probability Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  4. Programme 28: Probability Introduction Notation Types of probability

  5. Programme 28: Probability Introduction Notation Every trial or experiment has one or more possible outcomes. For example, rolling a six-sided die is a trial with six possible outcomes. An event, denoted by A, is a collection of one or more of those outcomes. For example, throwing an even number in the roll of a six-sided die is an event which consists of three outcomes. The probability, denoted by P(A) of an event A is a measure of the likelihood of the event occurring in any one trial or experiment. The probabilities of the various events associated with a trial can be defined in one of two ways.

  6. Programme 28: Probability Introduction Types of probability • The probabilities of the various events associated with a trial can be defined • either: • (a) Empirically: by repeating the experiment a number of times and noting the relative frequencies of the events. • (b) Classically: by defining the probabilities beforehand based on a knowledge of the experiment and its possible outcomes.

  7. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  8. Programme 28: Probability Empirical probability Expectation Success or failure Multiple samples Experiment

  9. Programme 28: Probability Empirical probability Empirical probability is based on previous known results. The relative frequency of the number of times an event has previously occurred is taken as the indication of its likely occurrence in the future For example: A random batch of 240 components is inspected and 20 are found to be defective. Therefore, if one component is picked at random from this batch, the chance of its being defective is 20 in 240. that is 1 in 12. In this case if A = {defective component} then the probability of selecting a defective component is:

  10. Programme 28: Probability Empirical probability Expectation The expectation E of an event A occurring in N trials is defined as the product of the probability of A occurring and the number of trials: For example, in a production run of 600 components where the probability of any one component being defective is 1/12 the expectation is that there will be: defectives in the run.

  11. Programme 28: Probability Empirical probability Success or failure If, in a trial, a particular event A does occur that can be recorded as a success. If the event does not occur that can be recorded as a failure. In N trials there will be x successes and N – x failures. Now: so that: Hence, since: then:

  12. Programme 28: Probability Empirical probability Success or failure Denoting by A the event {success} then notA is the event {failure} then: The event notA is denoted by the complement of A: That is:

  13. Programme 28: Probability Empirical probability Multiple samples A single sample of size n taken from a total population of size N is not necessarily representative of the population. Another sample of size n could well display different properties. However, continued sampling will provide a cumulative effect that does demonstrate consistency.

  14. Programme 28: Probability Empirical probability Experiment • Take a deck of 52 playing cards, shuffle well and deal out 12 cards (n) at random. • Count the number of spades (x) • Replace the sample, re-shuffle and repeat the process. • Calculate the average number of spades in the two samples – the running average. • In this way a table can be constructed:

  15. Programme 28: Probability Empirical probability Experiment The results from one such experiment gave the following distribution of spades: Number of spades in 40 trials of a 12-card sample

  16. Programme 28: Probability Empirical probability Experiment Plotting the running average (cum x)/r against r gave the following graph:

  17. Programme 28: Probability Empirical probability Experiment From the graph it is seen that the running average is settling down to a value of 3 spades in a sample of 12 cards. Since there are 13 spades in a deck of 52 playing cards there is an expectation that there will be: Spades in a sample of 12 – but this is taking us towards the second way of defining probabilities.

  18. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  19. Programme 28: Probability Classical probability The classical approach is to consider the number of ways that an event could possibly occur and take the ratio of that to the total number of possible outcomes: Since there are 13 spades in a deck of 52 playing cards, the probability of a card drawn at random being a spade is then:

  20. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  21. Programme 28: Probability Certain and impossible events Given that an event A can occur x times in n trials then: If event A is certain to occur then x = n, in which case: The probability of certainty is 1 and the probability of impossibility is 0. Therefore, for any event:

  22. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  23. Programme 28: Probability Mutually exclusive and mutually non-exclusive events Mutually exclusive events are events that cannot occur together. For example, in rolling a six-sided die a 5 and a 6 cannot be rolled at the same time in any one trial. Mutually non-exclusive events are events that can occur simultaneously. For example, in rolling a six-sided die a 6 and an even number can be rolled at the same time in any one trial

  24. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  25. Programme 28: Probability Addition laws of probability • If there are n possible outcomes to a trial, of which x give an event A and y • give an event B then if: • (a) A and B are mutually exclusive events, then: • (b) A and B are mutually non-exclusive events, then:

  26. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  27. Programme 28: Probability Independent events and dependent events Events are independent when the occurrence on one event does not affect the probability of the occurrence of the second event. For example, drawing one card from a deck and then drawing a second card after the first card has been replaced are independent events. Events are dependent when the occurrence on one event does affect the probability of the occurrence of the second event. For example, drawing one card from a deck and then drawing a second card after the first card has not been replaced are dependent events.

  28. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  29. Programme 28: Probability Multiplication law of probabilities If events A and B are independent events then:

  30. Programme 28: Probability PART 1 Introduction Empirical probability Classical probability Certain and impossible events Mutually exclusive and mutually non-exclusive events Addition laws of probability Independent events and dependent events Multiplication law of probabilities Conditional probability

  31. Programme 28: Probability Conditional probability The probability of event B occurring given that event A has already occurred is denoted by: If A and B are independent events the prior occurrence of event A will have no effect on the probability of the occurrence of B and so: If A and B are dependent events the prior occurrence of event A will have an effect on the probability of the occurrence of B and in this case:

  32. Programme 28: Probability PART 2 Discrete probability distribution Permutations and combinations General binomial distribution Mean and standard deviation of a probability distribution The Poisson probability distribution Continuous probability distributions Standard normal curve

  33. Programme 28: Probability PART 2 Discrete probability distribution Permutations and combinations General binomial distribution Mean and standard deviation of a probability distribution The Poisson probability distribution Continuous probability distributions Standard normal curve

  34. Programme 28: Probability Discrete probability distribution A coin is repeatedly tossed and the possible outcomes are listed: The results are then tabulated and the probabilities of tossing a head are listed:

  35. Programme 28: Probability Discrete probability distribution There is clearly a pattern here. The probabilities are the separate terms of the binomial expansion:

  36. Programme 28: Probability Discrete probability distribution Recall that the binomial coefficients are given by Pascal’s triangle: Further progress now requires a knowledge of permutations and combinations.

  37. Programme 28: Probability PART 2 Discrete probability distribution Permutations and combinations General binomial distribution Mean and standard deviation of a probability distribution The Poisson probability distribution Continuous probability distributions Standard normal curve

  38. Programme 28: Probability Permutations and combinations Permutations Combinations

  39. Programme 28: Probability Permutations and combinations Permutations Given the three letters A, B and C they can be arranged in 6 different ways: AB, AC, BC, BA, CA, CB Each arrangement of 2 letters is called a permutation and the permutations of 2 items out of 3 is denoted by: In this case it is seen that:

  40. Programme 28: Probability Permutations and combinations Permutations Given n different objects taken in arrangements of r at a time, the first one can be selected in n ways, the second in n – 1 ways, the third in n – 2 ways and so on until the rth can be selected in n – r + 1 ways to give: Now: That is:

  41. Programme 28: Probability Permutations and combinations Combinations Given the three letters A, B and C they can be selected in 3 different ways: AB, AC, BC (note that BA = AB; they are the same selection) Each selection of 2 letters is called a combination and the combinations of 2 out of 3 is denoted by: In this case it is seen that:

  42. Programme 28: Probability Permutations and combinations Combinations Given n different objects taken in selections of r at a time, the first one can be selected in n ways, the second in n – 1 ways, the third in n – 2 ways and so on until the rth can be selected in n – r + 1 ways to give: Each selection can be rearranged in r! different ways to give r! different permutations but still remain the same selection. Therefore there are: combinations.

  43. Programme 28: Probability PART 2 Discrete probability distribution Permutations and combinations General binomial distribution Mean and standard deviation of a probability distribution The Poisson probability distribution Continuous probability distributions Standard normal curve

  44. Programme 28: Probability General binomial distribution If a trial has two possible outcomes, success with a probability of p and failure with a probability of q then: p + q = 1 In a sequence of n trials the probability of r successes is given by the general binomial term: This distribution of probabilities is called the binomial probability distribution.

  45. Programme 28: Probability General binomial distribution For a binomial probability distribution: where n = 5, p = 0.2 and therefore q = 0.8 the following table can be constructed:

  46. Programme 28: Probability General binomial distribution • These values can then be displayed as a probability histogram: • The probability of any particular outcome is given by the height of each column, but since the columns are 1 unit wide this equates to the area of each column. • The total probability is 1 so: • the total area of the probability histogram is 1

  47. Programme 28: Probability PART 2 Discrete probability distribution Permutations and combinations General binomial distribution Mean and standard deviation of a probability distribution The Poisson probability distribution Continuous probability distributions Standard normal curve

  48. Programme 28: Probability Mean and standard deviation of a probability distribution There are two very simple and useful formulas for the mean and the standard deviation of a binomial probability distribution: where:

  49. Programme 28: Probability PART 2 Discrete probability distribution Permutations and combinations General binomial distribution Mean and standard deviation of a probability distribution The Poisson probability distribution Continuous probability distributions Standard normal curve

  50. Programme 28: Probability The Poisson probability distribution When the number of trials is large n≥ 50 and the probability of success is small p 0.1 the binomial probability distribution can be closely approximated by the Poisson probability distribution. For a sequence of n trials the Poisson probability of r successes is given as: where:

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