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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.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
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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 Principle P(E) = n(s) Product Rule P(EnF) = P(E) · P(FE) = P(F) · P(E|F) Union Rule For 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) Permutations P(n, k) Bayes' Theorem P(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 Probability n(EnF) P(ENF) P(F) n(F) Distinguishable Permutations n! n₁!n₂!...nk! P(E|F) = = Expected Value E(x) = x₁p₁ + x2P2 + ·+XnPn Binomial Probability P(k successes in n trials) = C(n. k)p(1-p)"-h ... Combinations C(n, k) = k!(n-k)! Expected # of Successes of a Binomial or Combination-based Random Variable E(x) = np = 2 Odds Odds in favor of E= If odds in favor of E are m:n, then P(E) m m+n Number of subsets If a set has n elements, then it has 2" subsets. = P(E) P(E') Independent Events P(E|F) = P(E) P(F|E) = P(F) P(EnF) = P(E). P(F) Table of Outcomes for Rolling 2 Dice 1 2 3 6 4 5 1|(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 Dice 2 3 4 5 6 5 6 6 7 8 1 1 2 3 4 2 3 4 5 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Irod
Transcribed Image Text:Basic Probability Principle P(E) = n(s) Product Rule P(EnF) = P(E) · P(FE) = P(F) · P(E|F) Union Rule For 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) Permutations P(n, k) Bayes' Theorem P(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 Probability n(EnF) P(ENF) P(F) n(F) Distinguishable Permutations n! n₁!n₂!...nk! P(E|F) = = Expected Value E(x) = x₁p₁ + x2P2 + ·+XnPn Binomial Probability P(k successes in n trials) = C(n. k)p(1-p)"-h ... Combinations C(n, k) = k!(n-k)! Expected # of Successes of a Binomial or Combination-based Random Variable E(x) = np = 2 Odds Odds in favor of E= If odds in favor of E are m:n, then P(E) m m+n Number of subsets If a set has n elements, then it has 2" subsets. = P(E) P(E') Independent Events P(E|F) = P(E) P(F|E) = P(F) P(EnF) = P(E). P(F) Table of Outcomes for Rolling 2 Dice 1 2 3 6 4 5 1|(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 Dice 2 3 4 5 6 5 6 6 7 8 1 1 2 3 4 2 3 4 5 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Irod
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