Probability Models and Simulation | SHUBLEKA
A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Example: Coin tossing (applet) 10,000 times Æ 5067 heads Æ proportion 0.5067 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Randomness: • • •
A long series of independent trials (the outcome of one trial must not influence the outcome of any other) Simulations start with given probabilities and imitate random behavior, we can only estimate realworld probability by actually observing many trials Simulations are very useful because we need long runs of trials. Short runs give only rough estimates. Simulation Æ Example TI-83
Probability Model: • • • •
A list of possible outcomes A probability for each outcome Sample space S = the set of all possible outcomes Event = is an outcome or a collection of outcomes. i.e. a subset of the sample space
Example: S = {Head, Tail} S = {0, 1, 2, 3, 4, 5} on phenomenon = toss four times and count heads Math Æ Prob Æ randInt(0, 9 , 1) creates 1 random integer between 0 and 9. randInt (0, 9, 5) Æ L1 creates and stores 5 random integers between 0 and 9 to List 1. Rules: 1. Any probability is a number between 0 and 1 2. All possible outcomes together must have probability = 1. 3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. 4. The probability that an event does not occur is 1 minus the probability that the event does occur. Next Æ Rules of Probability Simulation = an effective tool for finding likelihoods of complex results once we have a trustworthy model. The proportion of repetitions on which a result occurs will eventually be close to its true likelihood. Simulation problem: Is this discrimination? RandInt(1, 100, 10) ÆL1 : SortA(L1) : (L1 <=24)ÆL2:sum(L2)