Key Vocabulary: trial random probability independence random phenomenon sample space S = {H, T} tree diagram replacement event P(A) Complement A C disjoint Venn Diagram union (or) intersection (and) trial simulation joint event joint probability conditional probability Calculator Skills: randint( SortA Sum 1-Var Sats 5.1 Randomness, Probability, and Simulation 1. What does the Law of Large Numbers tell us? The law of large numbers says the proportion of times that a particular outcome occurs in many repetitions will approach a single number. 2. In statistics, what is meant by probability? The mathematics of chance; a probability of an event is always described as a value between 0 and 1, with 0 representing it is impossible for the event to occur and 1 representing it is guaranteed the event will occur. 3. What is wrong with the Law of Averages? It is a major misconception in probability for it suggests that outcomes will even out in a short number of trials.
4. What is simulation? Simulations are probability models used to imitate chance behavior or processes and to estimate probabilities. 5. List the four steps for conducting a simulation: STATE: Identify the question of interest about the chance process PLAN: Describe how to use a chance device to imitate a repetition of the chance behavior DO: Perform many repetitions of the simulation CONCLUDE: Use the results of the simulation to answer the question of interest 5.2 Probability Rules 1. In statistics, what is meant by an independent trial? The occurrence of the trial is not affected by any other trial 2. What is a sample space? A sample space includes all possible outcomes of a chance process. 3. What is an event? An event is a subset of the sample space and can be labeled as any collection of outcomes from some chance process. 4. Explain why the probability of any event is a number between 0 and 1. The probability of an event is the long-run proportion of repetitions on which that event occurs. Since any proportion is a number between 0 and 1, so are probabilities. 5. What is the sum of the probabilities of all possible outcomes? One
6. Describe the probability that an event does not occur? What is it called? The probability that an event does not occur is 1 minus the probability that the event does occur. It is called the complement of an event. 7. When are two events considered disjoint? What is another term for disjoint? Two events are considered disjointed when they have no outcomes in common. Another term for disjoint is mutually exclusive. 8. What is the probability of two mutually exclusive events? If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. 9. What is meant by the union of two or more events? Illustrate on a Venn diagram. The union (A B) of events A and B consists of all outcomes in event A, event B, or both. 10. State the addition rule for disjoint events. Illustrate on a Venn diagram. Addition rule for mutually exclusive events: If A and B are mutually exclusive, P(A or B) = P(A) + P(B).
11. State the general addition rule for unions of two events. The general addition rule can be used to find P(A or B): P(A or B) = P(A) + P(B) P(A and B) 12. Explain the difference between the rules in 10 and 11. The rule in number 10 is for independent or mutually exclusive events. The rule in number 11 is for non-independent of non-mutually exclusive events. 13. Summarize the 5 Rules of Probability. For any event A, 0 P(A) 1. If S is the sample space in a probability model, P(S) = 1. In the case of equally likely outcomes, number of outcomes corresponding to event A P(A) = total number of outcomes in sample space Complement rule: P(A C ) = 1 P(A) Addition rule for mutually exclusive events: If A and B are mutually exclusive, P(A or B) = P(A) + P(B). 14. What is meant by the intersection of two or more events? Illustrate on a Venn diagram. The intersection of events A and B (A B) is the set of all outcomes in both events A and B. 15. Explain the difference between the union and the intersection of two or more events. The union of two or more events includes all outcomes in the events. The intersection, however, only includes the outcomes that the events have in common.
5.3 Conditional Probability and Independence 1. What is meant by conditional probability? When we are trying to find the probability that one event will happen under the condition that some other event is already known to have occurred, we are trying to determine a conditional probability. 2. When are two events considered independent? State the formula used to determine if two events are independent? When knowledge that one event has happened does not change the likelihood that another event will happen, we say the two events are independent. P(A B) = P(A) and P(B A) = P(B). 3. State the general multiplication rule for any two events. The probability that events A and B both occur can be found using the general multiplication rule: P(A B) = P(A) P(B A) where P(B A) is the conditional probability that event B occurs given that event A has already occurred. 4. What is the multiplication rule for independent events? When events A and B are independent, we can simplify the general multiplication rule since P(B A) = P(B). Multiplication rule for independent events: If A and B are independent events, then the probability that A and B both occur is: P(A B) = P(A) P(B) 5. How is the general multiplication rule different than the multiplication rule for independent events? The general multiplication rule includes conditional probability. The multiplication rule for independent events cannot be conditional and independent; therefore, conditional probability is not included in the latter.
6. Can mutually exclusive events be independent? Two mutually exclusive events can never be independent simply because if one event happens, then the other event is guaranteed NOT to happen. 7. What is the opposite of at least one? None 8. State the formula for finding conditional probability. The conditional probability formula states: P(B A) = P(A B) / P(A). 9. Explain how to set up a tree diagram. Tree diagrams can be used to describe sample spaces, provided that the alternatives are not too numerous. Once the sample space is illustrated, the tree diagram can be used for determining probabilities. The probabilities for an event and NOT for an event are marked on the branches of the tree. All outcomes are represented and the probabilities of each outcome are obtained by multiplying the probabilities along that path. 10. What is meant by joint probability? Joint probability is used in multi-stage experiments when we want to find out how likely it is that two (or more) events happen at the same time. An example is drawing three cards and getting all jacks P(JJJ). We can use a tree diagram to figure out this type of probability. This type of probability depends on whether the experiment is done with or without replacement.