Introduction to Probability and Terminology Probability Topics 1
Introduction to Probability and Terminology Probability Topics 1
Probability � Study of randomness � Generally, � We probability means the chance of an event occurring. want to give CHANCE a definite, clear interpretation � Probability is the basis of statistical inference. 2
Describing Randomness � � � Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. Law of Large Numbers – as the number of repetitions of an experiment increases, the proportion with which a certain outcome is observed gets closer to the actual (classical) probability of that outcome. Ex. Toss a coin, we will never know the result in advance. But as you make more tosses, the proportion of heads will get closer and closer to 0. 5 3
Definitions � � � Probability Experiment – a repeatable process where the results are uncertain. Outcome – result of a single trial of a probability experiment. Sample Space– set of all possible outcomes of a probability experiment. Usually denoted by S. Event – an outcome or set of outcomes of a probability experiment. Probability Model – a mathematical description of a random phenomenon. Can take many forms but has two main parts: ◦ List of possible outcomes (sample space) ◦ A way of assigning probabilities to events. 4
Axioms of Probability A Probability is a number that is assigned to each member of a collection of events from a random experiment that satisfies the following properties: 1. P(S) = 1 2. 0 ≤ P(E) ≤ 1 3. For each two events E 1 and E 2 with E 1 ∩ E 2 = Ø, P(E 1 U E 2) = P(E 1) + P(E 2) (disjoint addition rule) 5
Approaches to finding probabilities � The Classical Approach � The Relative frequency approach � Subjective � Probability rules for more complicated situations
The Classical Approach � ◦ Classical Approach – uses sample spaces to determine the numerical probability that an event will happen (no actual experiment). It assumes that all outcomes in the sample space are equally likely to occur. die rolls, each # has prob. 1/6 � Probability of event E is � The probability of each outcome is 1/n where n is the number of outcomes in the sample space. 7
The Relative Frequency Approach - Uses frequency distributions based on observations to estimate probability. Relies on an actual experiment. We need not have all outcomes equally likely. Also called the empirical method. 8
Relative frequency Example � We roll a die 12 times and get a 3 twice. � We roll a die 12 times and get a 4 once. ◦ It appears the probability is 2/12 or 1/6 ◦ It would appear this probability is 1/12. � Is this correct? ◦ What could help up make sure our relative frequency probability is accurate?
Subjective Probability Personal (or Subjective) Probability - the degree to which a given individual believes the event in question will happen � Personal belief or judgment � Used to assign probabilities when it is not feasible to observe outcomes from a long series of trials. ◦ Assigned probabilities must follow established rules of probabilities (between 0 and 1…) 10
Law of Large Numbers Implications � If an experiment with a random outcome is repeated a large number of times, the relative frequency probability of an event is likely to be close to the true (Classical) probability. � The larger the number of repetitions, the closer together these probabilities are likely to be. � Streaks do not contradict the law and are much more common than most people believe. ◦ Ex. If you get a large number of heads in a row, then the next flip is more likely to come up tails.
The Compliment �
Relationships Between Events � In order to apply rules to find probabilities between two events we need to know their relationship � Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common. Therefore they cannot occur simultaneously. � Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. 13
Compound Events Sometimes we might want to look at combinations of events � The Union of two events is the event consisting of all outcomes that are contained in either of two events, denoted: A ⋃ B. Called A or B. � The Intersection of two events is the event consisting of all outcomes that contained in both of two events, A ⋂ B. Called A and B
Conditional Probability � Conditional probability quantifies the probability that a second event will occur, given that one has already occurred. � Denoted P(A∣B). Read probability of A given B. � Remember that P(A|B) and P(B|A) are NOT the same. 10
Defining Some Sample spaces � Sometimes want to write out an entire sample space if possible � Examples: o Coin toss o o Dice Roll o o {A♠, K♠, etc… }; 4 Suits (♠♣♥♦), 13 cards (2 -10, J, Q, K, A), for a total of 52 Two coins? o o {1, 2, 3, 4, 5, 6} Deck of cards o o {H, T} {HH, HT, TH, TT} Two dice? 16
Finding Probabilities � Consider � Each the Sample Space for Rolling 2 Die: individual outcome has probability 1/36 (. 0278) of occurring. � Using the classical approach find the following probabilities about the sum of two dice: � P(2) ◦ 1/36 � P(7) ◦ 6/36 � P(“Doubles”) ◦ 6/36 17
Probability Terminology Example Consider surveying all students in class on a given day. You record the results of the following questions. What do each of the questions represent? Questions Probabilities � Do you have any change in your pocket or purse? � P(change) => P(C) � Do you do NOT have change in your pocket? � P(change') => P(C’) � Did you ride the bus today. � P(bus) => P(B) The number that answered "yes" to Either the first OR second questions. � This is the Union. P(change OR bus) � � The number that answered "yes" to the first AND second questions. � Out of the students who answered "Yes" to Question 2, how many answered Yes to question 1? � P(C∪B) This is an Intersection. P(change AND bus) � P(C∩B) This a conditional probability. P(change|bus) � P(C|B) 18
Simple Compliment Rule Example �
“At Least 1” Example � If the probability of a product having no defects is 86%, what is the probability that a product has at least one defect?
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