Notes on Probability, Lecture notes of Statistics

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Counting - 1 1.3: Counting Textbook: 3.7 Objectives > Be familiar with counting rules/formulas (probability of the intersection of independent events, multiplicative rule) used in this course Be familiar with the equations for permutations and combinations and know when to use them > Be able to apply the counting rules/formulas in different probability calculations >» Be able to combine the counting rules with previously learned probability concepts (unions, intersections, etc.) Vv Motivation In many of the probability problems we've looked at so far, we’ve been able to easily list out all the elements in our sample space S and in some event of interest (let’s call it A). Example When rolling a fair six-sided die, let A be the event that we observe an even number. S$ = (1, 2, 3,4, 5, 6}; n(S) =6 A= {2, 4, 6} n(A) =3 Ss cand > a P(A)=2=2=05 = | s a In other situations, listing all the elements in S may be too tedious. However, we've learned how to compute the number of elements in S, n(S), and have seen scenarios where we can still list the number of elements in a particular event of interest A. We've then used this information to calculate probabilities. Example When rolling five six-sided dice, let A be the event that all five dice show the same number. n(S) = r™ = 65 = 7,776 A= (11111, 22222, 33333, 44444, 55555, 66666}; n(A) = 6 —_ = 0,0008 2776 P(A) = =