AP Stats Ch 6: Probability and simulation, the study of randomness

Before starting, you should know...

• have three different methods to calculate questions involving chance
• estimate likelihood by observing phenomenon many times and finding the relative frequency of results
• slow and inconvenient
• make a probability model and use it to answer questions
• might  not be doable based on what we know
• start with a model that reflects the phenomenon in some way, then imitate (simulate) the procedures for a number of repetitions
• quicker than repeating real phenomenon and results can be more accurate, and can answer more difficult questions
• can answer many kinds of problems
• probability: math that describes pattern of chance outcomes; can be used to infer things

Part 1: Simulation

• simulation: imitation of chance behavior based on model that accurately reflects phenomenon
• good for looking at likelihoods of complex results if we have good model
• can use random digits for simulation; here’s how
• know what phenomenon you are interested in
• assign random digits to outcomes so that each outcome’s chances to become picked are the same as the probability that the corresponding outcome will occur in nature
• ask for many of these random digits; that way, it is like we are repeating the phenomenon
• trials: another word for repetitions
• draw conclusions
• using simulations good for independent events, which means that the outcome of a trial does not affect the possibilities of the outcomes of the next trials
• Caution! Models are based on opinion and experience. Model must correctly describe phenomenon, or else simulation will be incorrect
• Simulating with calculator or computer
• can be used to conduct many repetitions
• just ask for it to produce a lot of random digits; these will be your repetitions

Part 2: Probability models

• probability: phenomena are random and relative frequencies of outcomes will settle down at a certain value after many trials
• few trials >>> unpredictable outcomes of events
• long run: predictable outcomes of events
• language we use in probability
• random: when a phenomenon’s outcomes are unpredictable but there is still a clear distribution of outcomes when there are a lot of trials
• probability: proportion of times a n outcome would occur when there are a lot of trials in a random phenomenon
• empirical: when something is based on observing, not theorizing (speculating, thinking, supposing)
• probability is empirical, because we must observe the outcomes before assigning a probability
• randomness
• probability describes random behavior; when we want to look at probability, we imagine what will happen if we do an infinite amount of trials
• for things to be to be random and to be able to be simulated, events must be independent
• again, remember that probability is empirical; we can estimate probability only by observing a lot of trials; simulating can help looking at the outcomes of the many trials easier
• uses of probability
• can be used in many fields, including science, measurements, life expectancy, etc.
• is important in chance studies in statistics
• probability models
• model for randomness
• will first list possible outcomes, then find probability for each outcome
• vocab involved
• sample space S: set of all possible outcomes of a phenomenon
• event: an outcome or a set of outcomes
• probability model: mathematical description of phenomenon containing sample space and a way of assigning probabilities of events
• Sample space S tells what are the outcomes and what makes up the outcomes
• each member is a possible sample
• methods to make sure you got all the possible outcomes
• tree diagram
• start with one point and then draw lines to all the possible outcomes in one event; then continue from the points of the event and draw lines from those points to the possible outcomes of the next event
• multiplication principle
• if you can do one event x ways and another event y ways, then the number of ways you can do both events is xy ways
• make a list of all the possible outcomes
• can be hard to do if sample space too large
• when we make selections from a number of choices
• selecting with replacement: after one trial, we place the selection back in with the choices before selecting again >>> the first trial’s outcome does not affect the outcomes of the following trials
• selecting without replacement: after one trial, we do not place the selection back in with the choices before selecting again >>> the first trial's outcome affects the outcomes of the following trials
• important to know whether the question is referring to selecting with or without replacement
• probability rules
• probability is between 0 and 1
• sum of all probabilities is 1
• if events are disjoint (aka mutually exclusive), or does not overlap, then that means they can’t happen simultaneously and P(A or B) = P(A) + P(B); called addition rule for disjoint events
• for jointed events (there are overlaps), P(A or B) = P(A) + P(B) - P(A and B)
• complement rule: complement of an event is that the probability that the event is not going to happen, or P(complement of A) = 1 - P(A is going to happen)
• P(AUB) means P(A or B), or P(A union B)
• P(AnB) means P(A and B), or P(A intersect B)
• circle with a slash through it means empty event, or there are no outcomes in the event
• if events are disjoint, then that means P(AnB) is empty; it is an empty event
• drawing a venn diagram is a great method to us
• to find probabilities of an event that is made up of several possibilities in a finite sample space, then assign probabilities to those outcomes, and add them together to get the probability of the event
• for equally likely outcomes
• in some events, we assume that outcomes are equally likely
• if this occurs, and if the event has k possible outcomes that are all equally likely, then P(one individual outcome) = 1/k, and P(A) = (number of outcomes in A)/(k)
• Independence and multiplication rule
• events are independent: knowing the outcome of one event does not change the probability of the outcomes of another event
• multiplication rule: if events are independent, P(AnB) = P(A) x P(B)
• works only if events are independent! can’t use if events are dependent!
• independence is usually assumed in a probability model if two events seem physically unrelated to each other
• Caution! Disjoint events are not independent!
• more applications of probability
• if events A and B are independent, then their complements are independent and the complement of one event is independent to another event
• multiplication rule can apply to more than two events

Part 3: General Probability Rules

• union: shown by a U-like sign; P(A union B) = P(AUB); means probability of __ or __
• can be used on more than two events
• for disjoint events: P(AUBU...) = P(A) + P(B) + ....
• for jointed events: subtract each overlapping event once
• P(AUB) = P(A) + P(B) - P(AnB)
• P(AnB): means P(A and B)
• disjointed: always (/), or an empty set; no outcomes are in P(AnB)
• jointed: look at the place that the two events overlap
• venn diagrams are really helpful for this
• if events are independent, P(AnB) = P(A) x P(B)
• there is also a method to calculate P(AnB), which will be discussed in the conditional probability section
• Conditional Probability: probability that an event will happen after another event has already occurred
• assigned probability of event can change if some other event has occurred
• shown by P(B|A)
• P(B|A) = P(AnB)/P(A)
• P(AnB) of any two events regardless of jointed or disjointed = P(A) x P(B|A)
• can also be written as P(B) x P(A|B) since P(AnB) = P(BnA)
• event B = the event whose probability we are interested in; event A = event that has already occurred
• extended multiplication rules
• intersection: probability that all the events listed will occur
• means as many events as necessary can be described
• use the multiplication rule; adjust it so it is similar to P(AnB) = P(A) x P(B|A)
• ex. P(AnBnC) = P(A) x (B|A) x P(C|(A and B))
• P(AnBnCnD) = P(A) x (B|A) x P(C|(A and B)) x P(D|(A and B and C))
• tree diagrams
• as shown below (as always, picture does not belong to me), multiplication can be extended to tree diagrams
• just multiply the probability of one event happening to the probability that the second event happens to get the probability that a scenario happens