1. A single die is rolled. Find the probabilities of the following results, and give your answers as reduced fractions. Show your work and justify your answers. Use the formula sheet to help if needed. a, The number rolled is less than 5, given that the number rolled is even. 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

The possible outcomes when rolling a die is {1,2,3,4,5,6}.

A single die is rolled. Find the probabilities of the following results, and give your answers as reduced fractions. Show your work and justify your answers. Use the formula sheet to help if needed. a, The number rolled is less than 5, given that the number rolled is even.

A single die is rolled. Find the probabilities of the following results, and give your answers as reduced fractions. Show your work and justify your answers. Use the formula sheet to help if needed. a, The number rolled is less than 5, given that the number rolled is even.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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