Interactive Solutions for All Problems - Chapters 1.2, 1.3, and 1.4
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Problem 1.2.1: Union and Intersection
Easy
Find the union \(C_1 \cup C_2\) and the intersection \(C_1 \cap C_2\) of the two sets \(C_1\) and \(C_2\).
Results
Problem 1.2.2: Set Complements
Easy
Find the complement \(C^c\) of the set \(C\) with respect to the space \(\mathcal{C}\).
Problem 1.2.3: Letter Arrangements
Medium
List all possible arrangements of the four letters m, a, r, and y. Let \(C_1\) be the collection of arrangements in which y is in the last position. Let \(C_2\) be the collection of arrangements in which m is in the first position. Find the union and intersection of \(C_1\) and \(C_2\).
Problems 1.2.5 through 1.2.17 are implemented below. Click on each to expand and interact.
Problem 1.3.1: Dice Probability
Easy
A die is cast. Compute probabilities for various events. Let \(C_1\) and \(C_2\) be subsets of the sample space \(\{1,2,3,4,5,6\}\).
Problem 1.3.3: Coin Toss Until Head
Medium
A coin is tossed until the first head appears. The sample space is H, TH, TTH, TTTH, ... with probabilities 1/2, 1/4, 1/8, 1/16, ... Show that \(P(\mathcal{C}) = 1\).
Problem 1.3.18: Poker - Full House Probability
Hard
Find the probability of being dealt a full house (three of one kind and two of another kind) in a 5-card poker hand.
Problem 1.4.30: The Monty Hall Problem
Hard
Three doors, one prize. You pick one, Monty opens another showing no prize. Should you switch doors?
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Choose a door!
Problem 1.4.8: Factory Machines - Bayes' Theorem
Medium
Three machines produce springs with different defect rates. Use Bayes' theorem to find probabilities.