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Probability Basics

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- 1. PROBABILITY BASICS 1. Introduction to Probability Generally, Probability deals with experiments which can have a number of different outcomes which then form the sample space. Sample Space - Is the set of all possible outcomes of the experiment. It is usually denoted by S. : If a person plants ten bean seeds and counts the number that germinate, the sample space is: S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. : If I toss a coin three times and record the result, the sample space is: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} : When two coins are tossed together the possible outcomes of the experiment are: HH, HT, TH and TT. So the sample space is given by: S={ } : Consider an experiment in which two dice are tossed. The sample space for this experiment is: S={ = {( )( } ) { ( )( } )( ) ( ) ( )( ) ( )} : If the outcome of an experiment consists in the determination of the sex of a Newborn child, then the sample spaces is: S={ } Where the outcome b means the child is a boy and g is a girl. Continuous sample space - if it contains an interval (either ﬁnite or inﬁnite) of real numbers. : Considering an experiment in which a molded plastic part is selected, such as a connector, and measure its thickness. The possible values for thickness depend on the resolution of the measuring instrument; they also depend on upper and lower bounds for thickness. So to deﬁne the sample space for this experiment can simply be the positive real line. { 1 }
- 2. Discrete sample space - if it consists of a ﬁnite or countable inﬁnite set of outcomes. : If it is known that all connectors will be between 10 and 11 millimeters thick, the sample space could be: { } : If the objective of the analysis is to consider only whether a particular part is low, medium, or high for thickness, the sample space might be taken to be the set of three outcomes: { } : If the objective of the analysis is to consider only whether or not a particular part conforms to the manufacturing speciﬁcations, the sample space might be simpliﬁed to the set of two outcomes: { } Event ( )– Is a subset of the sample space (S) of a random experiment. That is, an event is a set consisting of possible outcomes of the experiment. If the outcome of the experiment is contained in E, then it’s said that E has occurred. : In tossing a coin three times and recording the result, the sample space is: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} If there is a need to get two heads from the sample, then event is: E = {HHT, HTH, THH} where clearly . : In the outcome of an experiment consisting in the determination of the sex of a newborn child, the sample set is: S = {g, b} Where the outcome g means the child is a girl and b is a boy. If E = {g}, then E is the event that the child is a girl. If F = {b}, then F is the event that the child is a boy. 2
- 3. Complement of Event ( ) – Is the subset that consists of all outcomes in the sample space S that are not in the event. That is, will occur if and only if does not occur. In the outcome of an experiment consisting in the determination of the sex of a newborn child, in which the sample set, is: S = {g, b} 1. If { } is the event that the child is a boy, then 2. { } is the event that it is a girl. 3. Since the experiment must result in some outcome, it also follows that Random experiment without replacement – When items are not replaced before the next one is selected in the sample. Ex: Having a batch consisting of three items {a, b, c} and the experiment is to select two items without replacement; its sample space can be represented as: { } Random experiment with replacement – If items are replaced before the next one is selected when sampling. Ex: In a batch consisting of three items {a, b, c} and the experiment is to select two items with replacement, the possible ordered outcomes are: { } Exhaustive Cases – Is the total number of possible cases that can occur in an experiment. : In throwing a die, there are six exhaustive cases. Since any one of the faces marked 1, 2, 3, 4, 5 and 6 may come uppermost. : In tossing a coin there are two exhaustive cases head and tail since one of these can come uppermost. Favourable Cases – These are cases which ensure the occurrence of an event. 3
- 4. : In tossing a die, the number of cases Favourable to the appearance of multiple of 3 is two viz. 3 and 6. : In drawing two cards from a pack of 52 cards the number of cases favourable to drawing two aces in . Mutually Exclusive Cases (incompatible/disjoint) – If the happening of one case, prevents all other cases. So, for any sequence of mutually exclusive events ,… . : In tossing a coin the head and tail are mutually exclusive cases, since occurring of one (head/tail) prevents the other. Equally Likely Cases – When there is no any reason or condition to expect the happening of an event in preference of the other. : If a coin is tossed either head or tail occurs. There is no reason to expect that only head will be thrown. Getting head or tail in a coin toss are equally likely events. Definition of Probability: Probability theory – Is the study of random or unpredictable experiments. Given a random experiment, let n be the number of exhaustive, mutually exclusive and equally likely cases; and m cases out of them are favourable to happening of an event E, then the probability P(E) of happening of E is given by: ( ) If m cases are favourable to an event E, then n-m events are such, which are not favourable to E. = Therefore the probability of happening and not happening of the event are given by: ( ) Where always If ( ) ( ) ( ) ⟹ ( ) ( ) ( ) , E is called certain event and ( ) 4 is called an impossible event.
- 5. Odds in favour of event E ( ) Similarly Odds against event E ( ) If Odds in favour of event E are then ( ) If Odds again event E are then ( ) Assuming having n objects. The number of ways to fill n ordered slots with them is ( While the number of ways to fill k ( n ordered slots is ) ( ) ( ) ( ), for r ≤ n, Number of ways in which r objects can be selected from n ( ) ) ( ( ) ( ) ) The number of ways to divide a set of n elements into r (distinguishable) subsets of is denoted by ( elements where by: ( ( ) ) ( ) 5 ( ( ) ) ( ( ) ) ) is given

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