Math 106 Quiz 5 February 2013 Instructor S Sands

Math 106 Quiz 5february 2013 Instructor S Sandsname

Cleaned assignment instructions: The quiz is worth 100 points and comprises 6 problems. It is an open-book and open-notes exam. Students may refer to textbooks, notes, and online materials but must work independently without consulting others. All work or explanations must be shown where indicated, and submissions can be handwritten, typed, or scanned. The deadline for submission is February 24. The quiz covers probability, combinatorics, and statistics topics including sample spaces, events, probability calculations, expected value, Venn diagrams, and contingency tables.

Paper For Above instruction

The following academic paper comprehensively addresses the six problems outlined in the quiz, providing detailed solutions, explanations, and relevant statistical and probability concepts.

Analysis of Probability and Statistics Problems in Math 106 Quiz

Introduction

The quiz from Math 106 tests understanding of fundamental concepts in probability, combinatorics, and descriptive statistics. These problems encompass calculating sample spaces, events, probabilities, odds, combinatorial selections, expected values, Venn diagram analysis, and conditional probabilities in contingency tables. Each problem requires careful reasoning, application of formulas, and interpretation of results within the context of real-world scenarios.

Problem 1: Coin Purses and Probabilities

In the first problem, two coin purses are considered: the first with a nickel (N) and a quarter (Q), and the second with a dime (D) and a penny (P). The selection involves randomly choosing one coin from each purse, leading to a sample space of four possible outcomes: (N, D), (N, P), (Q, D), (Q, P).

For event A: "sum of coin values is even," outcomes with sums of 5, 27, 10, or 26 cents are analyzed. Only outcomes where the total sum of the two coins is even are listed (e.g., Q and D totaling 25 cents could be odd, so only to be included if total even).

Calculations for probabilities involve counting favorable outcomes over total outcomes, recognizing each equally likely outcome in the sample space.

Similarly, event B's outcomes include sums less than 10 or greater than 30 cents. Calculations involve identifying these outcomes, then computing probabilities based on total uniform outcomes. The intersection probability, P(A∩B), involves outcomes satisfying both conditions, with explanatory work based on counting and probability rules.

Problem 2: Odds Calculation

The probability of winning is 4/7, hence the odds in favor are "4 to 3" (successes to failures). Odds against are "3 to 4," reflecting the complement probability. This straightforward conversion from probability to odds illustrates an understanding of basic odds notation in sports and gambling contexts.

Problem 3: Combinatorics of Sandwich Selection

With 11 ham and 7 turkey sandwiches in a tray, 8 are randomly selected. Computing the probability of selecting exactly 3 ham and 5 turkey sandwiches involves hypergeometric distribution: the number of ways to choose 3 ham from 11, × number of ways to choose 5 turkey from 7, divided by total ways to choose 8 sandwiches from 18. The formula for hypergeometric probability is applied here, emphasizing combinatorial reasoning.

Problem 4: Expected Payoff and Fairness of a Game

The game involves multiple outcomes with associated probabilities and payoffs. The probability of winning some money (positive payoff) is calculated by summing probabilities where payoff > 0, i.e., winning $1, $4, or $5. The expected value is computed by summing the product of each payoff and its probability. The game’s fairness is determined by comparing the expected value to zero: if positive, the game favors the player; if zero, it is fair; if negative, unfavorable. This analysis relies on expected value formulas and probability-weighted payoffs.

Problem 5: Medical Study and Venn Diagram Analysis

The study on 100 patients examines the relief provided by drugs X and Y, with overlaps indicating patients who experienced relief from both. The data provide totals for relief by each drug and the overlap, facilitating the construction of a Venn diagram. The counts are used to find probabilities (e.g., P(X∪Y)), the probability that at least one drug relieves the migraine. Complementary probabilities assess cases where drugs fail or where relief occurs from only one medication. These involve set operations and probability rules for unions, intersections, and complements.

Problem 6: Blood Type Distribution and Conditional Probabilities

A contingency table displays blood types by sex in a sample of 600 individuals. The problem asks for probabilities based on marginal and joint distributions, including P(male), P(blood B), and conditional probabilities such as P(B|male). It further explores the independence of events M ("male") and B ("blood type B") by checking if P(M∩B) equals P(M)×P(B). Calculations involve basic probability formulas, proportion calculations, and independence criteria, which are central concepts in biostatistics and probability theory.

Conclusion

This comprehensive review and solution set demonstrate mastery of probability calculations, combinatorial reasoning, and statistical interpretation. Each problem reinforces core principles used extensively in various scientific and real-world applications, including risk assessment, decision making, and data analysis. Proper understanding and application of these foundational concepts are essential for success in advanced statistics and probability courses.

References

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