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Probability

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  1. Probability

  2. The definition – probability of an Event • Applies only to the special case when • The sample space has a finite no.of outcomes, and • Each outcome is equi-probable • If this is not true a more general definition of probability is required.

  3. Summary of the Rules of Probability

  4. The additive rule P[A B] = P[A] + P[B] – P[A  B] and if A  B = f P[A B] = P[A] + P[B]

  5. The Rule for complements for any event E

  6. Conditional probability

  7. The multiplicative rule of probability and if A and B areindependent. This is the definition of independent

  8. Counting techniques

  9. Summary of counting results Rule 1 n(A1 A2 A3  ….) = n(A1) + n(A2) + n(A3) + … if the sets A1, A2, A3, … are pairwise mutually exclusive (i.e. AiAj= f) Rule 2 N = n1n2 = the number of ways that two operations can be performed in sequence if n1 = the number of ways the first operation can be performed n2 = the number of ways the second operation can be performed once the first operation has been completed.

  10. N = n1n2 … nk = the number of ways the k operations can be performed in sequence if Rule 3 n1 = the number of ways the first operation can be performed ni = the number of ways the ith operation can be performed once the first (i - 1) operations have been completed. i = 2, 3, … , k

  11. Basic counting formulae • Orderings • Permutations The number of ways that you can choose k objects from n in a specific order • Combinations The number of ways that you can choose k objects from n (order of selection irrelevant)

  12. Applications to some counting problems • The trick is to use the basic counting formulae together with the Rules • We will illustrate this with examples • Counting problems are not easy. The more practice better the techniques

  13. Random Variables Numerical Quantities whose values are determine by the outcome of a random experiment

  14. Random variables are either • Discrete • Integer valued • The set of possible values for X are integers • Continuous • The set of possible values for X are all real numbers • Range over a continuum.

  15. Examples • Discrete • A die is rolled and X = number of spots showing on the upper face. • Two dice are rolled and X = Total number of spots showing on the two upper faces. • A coin is tossed n = 100 times and X = number of times the coin toss resulted in a head. • We observe X, the number of hurricanes in the Carribean from April 1 to September 30 for a given year

  16. Examples • Continuous • A person is selected at random from a population and X = weight of that individual. • A patient who has received who has revieved a kidney transplant is measured for his serum creatinine level, X, 7 days after transplant. • A sample of n = 100 individuals are selected at random from a population (i.e. all samples of n = 100 have the same probability of being selected) . X = the average weight of the 100 individuals.

  17. The Probability distribution of A random variable A Mathematical description of the possible values of the random variable together with the probabilities of those values

  18. The probability distribution of a discrete random variable is describe by its : probability functionp(x). p(x) = the probability that X takes on the value x. This can be given in either a tabular form or in the form of an equation. It can also be displayed in a graph.

  19. Example 1 • Discrete • A die is rolled and X = number of spots showing on the upper face. • formula • p(x) = 1/6 if x = 1, 2, 3, 4, 5, 6

  20. Graphs To plot a graph of p(x), draw bars of height p(x) above each value of x. Rolling a die

  21. Example 2 • Two dice are rolled and X = Total number of spots showing on the two upper faces. Formula:

  22. Rolling two dice

  23. 36 possible outcome for rolling two dice

  24. Comments: 1. The probability assigned to each value of the random variable must be between 0 and 1, inclusive: 2. The sum of the probabilities assigned to all the values of the random variable must equal 1: 3. Every probability function must satisfy:

  25. Example P ( the random variable X equals 2 ) 3 = = p ( 2 ) 14 In baseball the number of individuals, X, on base when a home run is hit ranges in value from 0 to 3. The probability distribution is known and is given below: Note: • This chart implies the only values x takes on are 0, 1, 2, and 3. • If the random variable X is observed repeatedly the probabilities, p(x), represents the proportion times the value x appears in that sequence.

  26. A Bar Graph

  27. Discrete Random Variables Discrete Random Variable: A random variable usually assuming an integer value. • a discrete random variable assumes values that are isolated points along the real line. That is neighbouring values are not “possible values” for a discrete random variable Note: Usually associated with counting • The number of times a head occurs in 10 tosses of a coin • The number of auto accidents occurring on a weekend • The size of a family

  28. Continuous Random Variables Continuous Random Variable: A quantitative random variable that can vary over a continuum • A continuous random variable can assume any value along a line interval, including every possible value between any two points on the line Note: Usually associated with a measurement • Blood Pressure • Weight gain • Height

  29. Random Variables Numerical Quantities whose values are determine by the outcome of a random experiment

  30. The probability distribution of a discrete random variable The probability distribution of a discrete random variable is describe by its : probability functionp(x). p(x) = the probability that X takes on the value x. This can be given in either a tabular form or in the form of an equation. It can also be displayed in a graph.

  31. Example

  32. Probability Distributionsof Continuous Random Variables

  33. Probability Density Function The probability distribution of a continuousrandom variable is describe by probability density curve f(x).

  34. Notes: • The Total Area under the probability density curve is 1. • The Area under the probability density curve is from a to b is P[a < X < b].

  35. Normal Probability Distributions(Bell shaped curve)

  36. Mean and Variance (standard deviation) of aDiscrete Probability Distribution • Describe the center and spread of a probability distribution • The mean (denoted by greek letter m (mu)), measures the centre of the distribution. • The variance (s2) and the standard deviation (s) measure the spread of the distribution. • s is the greek letter for s.

  37. Mean, Variance (and standard deviation) of aProbability Distribution

  38. Mean of a Discrete Random Variable • The mean, m, of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together: • Notes: • The mean is a weighted average of the values of X. • The mean is the long-run average value of the random variable. • The mean is centre of gravity of the probability distribution of the random variable

  39. Variance and Standard Deviation 2 s = s Variance of a Discrete Random Variable: Variance, s2, of a discrete random variable x is found by multiplying each possible value of the squared deviation from the mean, (x-m)2, by its own probability and then adding all the products together: Standard Deviation of a Discrete Random Variable: The positive square root of the variance:

  40. Example The number of individuals, X, on base when a home run is hit ranges in value from 0 to 3.

  41. Computing the mean: • Note: • 0.929 is the long-run average value of the random variable • 0.929 is the centre of gravity value of the probability distribution of the random variable

  42. Computing the variance: • Computing the standard deviation:

  43. Random Variables Numerical Quantities whose values are determine by the outcome of a random experiment

  44. Random variables are either • Discrete • Integer valued • The set of possible values for X are integers • Continuous • The set of possible values for X are all real numbers • Range over a continuum.

  45. The Probability distribution of A random variable A Mathematical description of the possible values of the random variable together with the probabilities of those values

  46. The probability distribution of a discrete random variable is describe by its : probability functionp(x). p(x) = the probability that X takes on the value x. This can be given in either a tabular form or in the form of an equation. It can also be displayed in a graph.

  47. Example P ( the random variable X equals 2 ) 3 = = p ( 2 ) 14 In baseball the number of individuals, X, on base when a home run is hit ranges in value from 0 to 3. The probability distribution is known and is given below: Note: • This chart implies the only values x takes on are 0, 1, 2, and 3. • If the random variable X is observed repeatedly the probabilities, p(x), represents the proportion times the value x appears in that sequence.

  48. A Bar Graph

  49. Probability Distributionsof Continuous Random Variables