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BUSN2429 Chapter 4: Intro to Probabilities P(A) x P(B/A) = P(B) x P(A/B) o P(B/A) = P(B) x P(A/B) / P(A) 4.1 Introduction to Probabilities Probability – is a numerical value ranging from 0 to 1 o Probability indicates the chance, or likelihood, of a specific event occurring If there is no chance of the event occurring, the probability is 0 Experiment –^ ^ If the event is absolutely going to occur, the probability of it occurring is the process of measuring or observing an activity for the purpose of^^1 collecting data ^ - Sample space –^ E.g.,^ rolling a single six-side die all the possible outcomes, or results, of an experiment
- The sample space for our single-die experiment is (1,2,3,4,5,6) Event – o The outcome, or outcomes, is a subset of the sample space one or more outcomes of an experiment - E.g., rolling a pair with 2 dice Simple event – an event with a single outcome in its most basic form that cannot be simplified o E.g., rolling a five with a single die
Methods of assigning probability: 1. Classical probability – used when the number of possible outcomes of the event of interest is known P(A) = # of possible outcomes that constitute Event A / Total # of possible outcomes in the sample space o E.g ., rolling a 5 , P(A) = 1/6 = 0.167 = 16.7% probability This is a Simple Probability : it represents the likelihood of a single Classical probability assumes that each event in the sample space has the^ (simple) event occurring by itself same likelihood of occurring The set of events is possible event that can occur collectively exhaustive - if the sample space includes every
2. Empirical Probability with which an event occurs – involves conducting an experiment to observe the frequency P(A) = Frequency in which event A occurs / Total # of observations
Law of large numbers – states that when an experiment is conducted a large number of times, the empirical probabilities of the process will converge to the classical probabilities o E. g., flip a coin a large number of times – the observed number of heads would be very close to 50%
3. Subjective Probability – available used when classical and empirical probabilities are not Instead use experience or intuition to estimate the probabilities E.g., the probability that inflation will be greater than 4% next year Basic properties of a probability: Probability Rule 1 – if P(A) = 1, then with certainty, Event A must occur Probability Rule 2 – if P(A) = 0, then with certainty, Event A will not occur Probability Rule 3 –Probability Rule 4 – The probability of any event must range from 0 to 1the sum of all the probabilities for the simple events in the sample size must be equal to 1 Probability Rule 5 – the sample space that are not part of the Event A the complement to Event A is defined as all of the outcomes in The complement is denoted as A’: Formula = P(A) + P(A’) = 1 or P(A) = 1 – P(A’) 4.2 Probability Rules for More than One Event Contingency Tables Contingency table – shows the number of occurrences of events that are classified according to two categorical variables Event A = a student living in the dorm being a female Event B = a student living in the dorm being a freshman
The probability that Event A, or Event B, or both events will occur Mutually exclusive – if they cannot occur at the same time during the experiment
For mutually exclusive events, the addition rule states that the probability of two events occurring is simply the sum of their individual probabilities:
If Events A and B are not mutually exclusive:
P(A and B) = 0 if events A and B are mutually exclusive Conditional Probability Conditional Probability – Event B has occurred is the probability of Event A occurring, given the condition that Also known as a posterior probability – a revision of the prior probability using additional information
Independent and Dependent Events Two events are considered has no impact on the occurrence of the other event independent of one another if the occurrence of one event
P(A or B) = P(A) + P(B) P(A or B) = P(A) + P(B) - P(A and B)
If the occurrence of one event affects the occurrence of another event, the events are considered dependent Formula for determining if Events A and B are independent: P(A|B) = P(A) If P(A|B) ≠ P(A), then events A and B are dependent
The Multiplication Rule Multiplication rule – probability) of two events occurring, or P(A and B) used to determine the probability of the intersection (joint Formula for the multiplication rule for dependent events:
or
Formula for the multiplication rule for two independent events: o P(A and B) = P(A)P(B) o Since events A and B are independent, P(B|A) = P(B) When multiple events are all independent, the probability of them all occurring is simply the product of their individual probabilities:
Contingency Tables with Probabilities
Formula for Bayes’ Theorem for n events:
o Ai = The i th^ event of interest from a choice of n events o B = An event that has already occurred
4.3 Counting Principles The fundamental counting principle states that: if there are kk 1 choices for the first event, k^2 choices for the second event... on choices for the nthen the total number of possible outcomes are:th^ event (k E.g., suppose a 3-digit model number is used to be selected using an initial A,B,C,D;^1 )(k^2 )(k^3 )…(kn) followed by two numbers from 0 – 9. How many different model numbers are possible o Answer: There are four letter choices, 10 number choices for the first and second number, so the number of possible outcomes is (4)(10)(10) = 400 Permutations
Permutations – order are the number of different ways in which objects can be arranged in The number of permutations of n distinct objects is n! : n! = n(n-1)(n-2)…(2)(1) o 0! = 1 Formula for the permutations of n objects selected x at a time: o n = the total number of objects o x = the number of objects to be selected E.g., suppose that 2 letters are to be selected from A,B,C,D and arranged in order. How many permutations are possible o Answer: The number of permutations, with n = 4 and x = 2, is o The permutations are: AB AC AD BA BC BD CA CB CD DA DB DC Combinations Combinations – are the number of different ways in which objects can be arranged without regard to order Formula for the combinations of n objects selected x at a time:
o When the order of objects is important, use permutations. When the order of objects does not matter, use combinations E.g. are possible (i.e., order is not important)?, Suppose that 2 letters are to be selected from A,B,C,D. How many combinations o Answer: the number of combinations is 6
o The combinations are: AB (same as BA), AC (same as CA), AD (same as DA), BC (same as CB), BD (same as DB), CD (same as DC)
