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7.3 Introduction to Probability The main concept for probability is that of event. Events continue the ideas of sets, but have some different terminology: Correlation of set terms vs. probability terms –(no corollary for sets!) experiment (doing something & keeping track of something about the result(s)) –Universal set  sample space (“certain event”, “sure event”) The sample space is essentially a list of all possible results that could be recorded for whatever the experiment is. –Element  outcome (also: sample point, elementary outcome, simple event) any one of the possible results –Set  event (any subset of the sample space!) –Empty set  null event (“impossible event”)

7.3 Introduction to Probability –Operations : Essentially the same for events as for sets – union A  B for sets E  F or “E or F” for events – intersection A  B for sets  E  F or “E and F” for events – complement A for sets  E or “not E” for events –Note the use of words as an alternative to set symbols! or and not used like before! –Disjoint sets  mutually exclusive events No elements or outcomes in common

7.3 Introduction to Probability Example: A fair coin is tossed. The outcome of the toss is recorded, as far as whether it was a “head” or a “tail”. –The sample space is a list of all possible results (the various outcomes which could be recorded). Here S = {H,T} –The H and the T are the 2 sample points (or elementary events, or simple outcomes) of the experiment, both equally-likely. –Any subset of S is an event. So here, the possible events are {} (the empty event), {H}, {T}, and {H,T} (the original S), so this experiment has 4 different events. –Probabilities are: P({})=0 (it’s impossible!), P({H})=1/2, P({T})=1/2, P({H,T})=1 (certain!)

7.3 Basic probability principles Formula: P(E) = n(E) / n(S) –if all events in S are equally likely –involves counting the size of sets Uniform sample space –means all outcomes are equally likely “equally likely” a must! –examples with a die –examples with two dice (On board)

7.3 Basic probability principles Example: A single fair die is tossed once and the uppermost face is recorded. Here the sample space S = {1,2,3,4,5,6} Some possible events: –A four is obtained: E 1 = {4} (this is a simple event) –An even # is obtained: E 2 = {2,4,6} –A # larger than 2 is obtained: E 3 = {3,4,5,6} –None of these six #’s is obtained: E 4 = {} (impossible!) How many different events are there?

–E 1 = {4} –E 2 = {2,4,6} –E 3 = {3,4,5,6} –E 4 = {} How many different events are there? (Events – think “subsets”) Since n(S)=6, the size of the power set is 2 6 =64, so there are 64 different events (including the null event). The above are only 4 of the 64 such events Note carefully: 6 different outcomes, 64 different events! 7.3 Basic probability principles

To calculate probabilities for these: –Keep in mind that S = {1,2,3,4,5,6} and n(S) = 6 and that S is uniform Probability of E 1, i.e., that a four is obtained: –We could write this as P(a “four” is obtained) or P(E 1 ) or P( {4} ) or P(4), as the simplest way (though somewhat informal) –To calculate the probability –P(E 1 ) = n(E 1 ) / n(S) – = 1 / 6 7.3 Basic probability principles

In a similar manner, P(even #) = P(E 2 ) = P({2,4,6})= n(E 2 ) / n(S) = 3 / 6 = 1/2 or.5 or 50% P(# larger than 2) = P(E 3 )= P({3,4,5,6}) = n(E 2 ) / n(S) = 4 / 6 = 2/3 P(none of these 6 #s) = P(E 4 )= P({}) = n({}) / n(S) = 0 / 6 = 0 Note that the size of the power set, in other words, the number of all possible events, is not used in these calculations. –Size of power set was 64 7.3 Basic probability principles

Example: Two fair fair dice are tossed once and the sum of the uppermost faces is recorded. The sample space S = {2,3,4,5,6,7,8,9,10,11,12} Some possible events: –A four is obtained: E 1 = {4} (this is a simple event) –An even # is obtained: E 2 = {2,4,6,8,10,12} –A # larger than 2 is obtained: E 3 = {3,4,5,…,12} –None of these eleven #’s is obtained: E 4 = {} (impossible!) How many different events are there? –2 11 = 2048 Note carefully: 11 different outcomes, 2048 different events! 7.3 Basic probability principles

For any event E, 0  n(E)  n(S) dividing by n(S) gives 0 / n(S)  n(E) / n(S)  n(S) / n(S) so that 0  P(E)  1 7.3 Basic probability principles

For any event E, 0  P(E)  1 –P.I.N.G.T.O. ! Probability is never greater than one! –P.I.N.N. ! Probability is never negative! Quantitative concepts (possible values for prob.?) – impossible0 – exceedingly unlikely.0001 ? – unlikely.2,.3 ? – fifty-fifty.5 ? – somewhat likely.7 ? – very likely.8,.9 ? – extremely likely.99 ? – sure (or certain)1 7.3 Basic probability principles