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Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help. a. Both dice are ≥4, given that the sum is ≥9.

Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help. a. Both dice are ≥4, given that the sum is ≥9.

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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2. Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help.

a. Both dice are ≥4, given that the sum is ≥9. 

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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