2. Two dice are rolled. Find the probabilities of the following as reduced fractions and justify. Use the formula sheet to help you out. a. Are the events "first die is even" and "sum = 10" independent? Basic Probability PrincipleP(E) = n(s)Product RuleP(EnF) = P(E) · P(FE) = P(F) · P(E|F)Union RuleFor sets: n(AUB) = n(A) + n(B) - n(ANB)For probability: P(EUF) = P(E) + P(F) − P(E^F)Complement Rule (Probability)P(E') = 1 - P(E)P(E) = 1 - P(E')Complement Rule (Number of Outcomes)n(E) = n(S) - n(E')n(E') = n(S) — n(E)PermutationsP(n, k)Bayes' TheoremP(F;).P(E|F;)P(F;|E) = P(F1).P(E|F1) + P(F2)·P(E|F2) + ... + P(Fn)·P(E|Fn)P(F).P(EF)Special Case: P(F|E) = P(F).P(E|F)+P(F').P(E\F')n!(n-k)!Conditional Probabilityn(EnF) P(ENF)P(F)n(F)Distinguishable Permutationsn!n₁!n₂!...nk!P(E|F) ==Expected ValueE(x) = x₁p₁ + x2P2 +·+XnPnBinomial ProbabilityP(k successes in n trials) = C(n. k)p(1-p)"-h...CombinationsC(n, k) = k!(n-k)!Expected # of Successes of a Binomialor Combination-based Random VariableE(x) = np=2OddsOdds in favor of E=If odds in favor of E are m:n,then P(E)mm+nNumber of subsetsIf a set has n elements,then it has 2" subsets.=P(E)P(E')Independent EventsP(E|F) = P(E)P(F|E) = P(F)P(EnF) = P(E). P(F)Table of Outcomes for Rolling 2 Dice12364 51|(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)5 (5,1) (5,2) (5.3) (5,4) (5,5) (5,6)6(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)Table of Sums when Rolling 2 Dice2 3 4 5 65 66 7 811 2 3 423 453 4 567 8 94 5 678 9 105 6 789 10 116 7 89 10 11 12Irod

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2. Two dice are rolled. Find the probabilities of the following as reduced fractions and justify. Use the formula sheet to help you out. a. Are the events "first die is even" and "sum = 10" independent?

2. Two dice are rolled. Find the probabilities of the following as reduced fractions and justify. Use the formula sheet to help you out. a. Are the events "first die is even" and "sum = 10" independent?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 6E

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