Chapter 5: ProbabilityRandomness•A random phenomenon is a situation in which we know what outcomes could happen, but we don't know which particular outcome did or will happen•However, we can calculate the probability with which each outcome will happen•People are not good at identifying truly random samples or random experiments, so we need to rely on outside mechanisms such as coin flips or random number tablesSimulating Randomness•Flip a coin to generate random 0's and 1’s•Pick a card to generate random numbers•Pick a number out of a hat•Use a Random Number table...Simulating Randomness•Use a Random Number table•Ex. Simulate rolling a die 10 times•Pick any line to start, say line 30 to start•Go through the numbers, picking numbers 1-6 and ignoring 0,7,8,9.•The“random”numbers are: 5, 4, 5, 3, 4, 6, 2, 5, 3, 116% of cars fail pollution tests in CaliforniaWe can represent this in many ways:•In a bag with 100 chips, 16 are red and represent cars that fail pollution tests, the remaining 84 are black and represent cars that pass the tests.•In a bag with 50 marbles, 8 are orange and represent cars that fail pollution tests, the remaining 42 are blue and represent cars that pass the tests.•On a random number table numbers 00-15 represent cars that fail pollution tests and 16-99 represent cars that pass the tests.Estimating Probabilities via SimulationWe would like to estimate the probability that an entire fleet of seven cars would pass using a random number table. We assume each car is independent.•Start at row 1 of the random number table and read two digits at a time.•If the random number is between 00-15, the car will fail the pollution test.•If the number is between 16-99, the car will pass the test.•A fleet of cars is comprised of seven cars, i.e. seven 2-digit random numbers. If all seven cars pass the test, record a 1, if not record a 0.•Repeat many times and calculate the proportion 1s among the total number of runs, i.e. number of fleets where all cars passed the pollution test.Run 1Run 2Run 3Did all the cars pass in run 3?NoRun 4Estimation of ProbabilityBased on the simulation results, estimate the probability that an entire fleet of seven cars would pass the pollution test.1/4=0.25 probability all 7 cars passNotes•4 runs is usually not considered sufficient, for homework use at least 5 runs.•You can start at any row you like on the random number table, but you should make sure to note it when you’re writing up your simulation scheme.•You should not arbitrarily pick numbers from the random number table. Just pick a row and follow across. Otherwise the numbers you’re using won’t be random (they will instead be your choices).Estimation of Probability•Based on the simulation results, estimate the average number of cars that fail the pollution test in a fleet of seven cars. a) 0 b) 1 c) 2 d) 3 e) 4Randomness•A random phenomenon is a situation in which we know what outcomes could happen, but we don't know which particular outcome did or will happen•However, we can calculate the probability with which each outcome will happenRandomness to Probability•Let's say there are 1000 songs in your iTunes library. Only 5 of these songs are by Justin Bieber. What is the probability that next time you hit shuffle you get a Justin Bieber song?•Prob(JB song) = 5/1000 = 0.005Definitions of Probability•A Frequentist: The probability of an event is its long-run relative frequency.•Theoretical: We may not be able to predict which song we play each time we hit shuffle, however we know that in the long run 5 out of 1000 songs will be Justin Bieber songs.•Empirical: We listen to 100 songs on shuffle and 4 of them are Justin Bieber songs. The empirical probability is 4/100=0.04•A Bayesian: •Prior probability of an event is the best guess by the observer of an event's probability. •Posterior probability of an event is the probability of an event after collecting some empirical data. It is obtained by updating information from the prior probability with additional data related to the event in question. The prior probability may be subjective, but with enough data, has little impact on the posterior probability.what you actually saw happenDefinitions of ProbabilityNote: In this course we will use the frequentist definition of probability, though we will make some use of Bayesian tools.More Definitions•For any random phenomenon, each attempt is called a trial and each trial generates an outcome•Each time you hit shuffle is a trial.•The song that plays as a result of hitting shuffle is an outcome.•Sample space is the collection of all possible outcomes of a trial•Sample space is the entire iTunes library of 1000 songs.•A combination of outcomes is called an event•For example playing two Justin Bieber songs in a row in shuffleSample Space•A couple has two kids, what is the sample space for the gender of these kids?Calculating ProbabilitiesIf a couple has two kids what is the probability that both are boys? If a couple has two kids, the list of possible scenarios are: BB, GG, BG, GB Since each scenario is equally likely: Prob(BB) = ¼ = 0.25Independence•When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. •Roughly speaking, this means that the outcome of one trial doesn't influence or change the outcome of another.•If the iTunes shuffle is truly random then the songs played are independent of each other.•In other words, the iTunes shuffle is memoryless, it does not say to itself “Wait, I just played an Justin Bieber song, I shouldn't play one again”•Similarly, if the genders of kids a couple has are independent, then the probability of having a boy for a second child doesn't change based on whether or not the first child of the couple was a girl.Definitions - Coin Toss Example•Trial: Each coin toss•Outcome: Heads or Tails•Probability: P(H) = 0.5 and P(T) = 0.5Note: Each time we toss a coin we can't tell which side will come up, however in the long run tails will come up 50% of the time and heads will come up 50% of the time•Sample Space: Tossed once: S = {H,T} Tossed twice: S = {HT,TH,HH,TT}•Independence: The outcome of one coin toss does not affect the outcome of the