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6.2 Introduction to Probability Part 2

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6.2 Introduction to Probability Part 2 6.2 Introduction to Probability part 2 ! From Tables to Probability " Computing relative frequency probabilities from a table. ! Probability Distributions 1 Summary Table ! The table below provides information on the hair color and eye color of 592 statistics students (The American Statistician, 1974). Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592 2 Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592 ! If a student is drawn at random from this population… " What is the probability that they are blond? ! ANS: 127/592 = 0.2145 " What is the probability that they are NOT blond? ! ANS: P(NOT blond) = 1-P(blond) =1-127/592 = 0.7855 ! ANS: (71 +286 + 108)/592 = 465/592 = 0.7855 3 Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592 ! If a student is drawn at random from this population… " What is the probability that they are blond and have brown eyes? ! ANS: 7/592 = 0.0118 " What is the probability that they have red hair and have green eyes? ! ANS: 14/592 = 0.0236 4 Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592 ! Based on this table… " What is the probability that a blond student has brown eyes? We’ve subsetted to a smaller ! ANS: 7/127 = 0.0551 population from which we are drawing a person " What is the probability that a brown-haired student has hazel eyes? ! ANS: 54/286 = 0.1888 5 Relative Frequency Method ! Step 1. Repeat or observe a process many times and count the number of times the event of interest, A, occurs. ! Step 2. Estimate P(A) by number of times A occurred ! P(A) = total number of observations 6 Example: 500-year flood ! Geological records indicate that a river has crested above a particular high flood level four times in the past 2,000 years. What is the relative frequency probability that the river will crest above the high flood level next year? number of years with flood = 4 = 1 total number of years 2000 500 The probability of having a flood of this magnitude in any given year is 1/500, or 0.002. 7 Probability Distributions ! A probability distribution shows the possible values that could occur, and the probability for each occurring. The probability distribution for the results of tossing two coins. 8 Probability Distributions ! A probability distribution shows the possible values that could occur, and the probability for each occurring. The possible values can be seen on the horizontal axis. 9 Probability Distributions ! A probability distribution shows the possible values that could occur, and the probability for each occurring. The probability of each value is shown by the height of the bar. 10 Example: Roll two 6-sided dice ! Find the probability distribution for the ‘sum of two dice’. Most likely sum is a sum of 7. 0.18 0.16 0.14 0.12 0.10 0.08 Only one way 0.06 to get sum of Probability 0.04 12 (double 6’s). 0.02 P(sum of 12) = 2 3 4 5 6 7 8 9 10 11 12 1/36 = 0.02778 Sum of two dice 11 Example: Roll two 6-sided dice ! Find the probability distribution for the ‘sum of two dice’. 0.18 0.16 0.14 0.12 Recall, 0.10 P(sum of 11) = 0.08 2/36 = 0.0556 0.06 And that is the Probability 0.04 height of the bar 0.02 above the number “11”. 2 3 4 5 6 7 8 9 10 11 12 Sum of two dice 12 Example: Roll one 6-sided dice ! Find the probability distribution for ‘the number that turns up’. 1.0 0.8 0.6 probability 0.4 Each of 6 values is equally likely, at 1/6 0.2 = 0.1667 0.0 1 2 3 4 5 6 value on die This is a uniform distribution. 13 Probability Distributions ! How to make a probability distribution. 1) List all possible outcomes. Use a table or figure it is helpful. 2) Identify outcomes that represent the same event. Find the probability of each event. 3) Make a table in which one column lists each event and another column lists each probability. The sum of all the probabilities must be 1. 14 Example: Making a probability Probability Distribution ! Random experiment: Toss Three Coins ! Make a probability distribution for the number of heads that occurs. 1) List the possible outcomes (equally likely): HHH,HHT,HTH,THH,HTT,THT,TTH,TTT 2) Identify outcomes that represent the same event. There are four possible events: 0, 1, 2, or 3 heads. 15 Example: Making a probability Probability Distribution ! Random experiment: Toss Three Coins ! Make a probability distribution for the number of heads that occurs. Number of heads Probability 3) Make a table 0 1/8 0: TTT (1 outcome) 1 3/8 1: HTT,THT,TTH (3 outcomes) 2 3/8 2: HHT,HTH,THH (3 outcomes) 3 1/8 3: HHH (1 outcome) This column sums to 1. 16 .
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