A Fair Coin Is Flipped 9 Times What Is The Probability Of Ge

1 A Fair Coin Is Flipped 9 Times What Is The Probability Of Getting

Calculate the probability for each of the following scenarios involving coin flips, marbles, dice, and other stochastic processes as specified:

1. A fair coin is flipped 9 times. What is the probability of getting exactly 6 heads? A fair coin means it has a 50% chance of landing heads up and a 50% chance of landing tails up.

2. You flip a coin three times. (a) What is the probability of getting heads on only one of your flips? (b) What is the probability of getting heads on at least one flip?

3. A jar contains 10 blue marbles, 5 red marbles, 4 green marbles, and 1 yellow marble. Two marbles are chosen without replacement. (a) What is the probability that one will be green and the other red? (b) What is the probability that one will be blue and the other yellow?

4. A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangoes. (a) What is the probability of pulling out a pear? (b) If you replace the first piece of fruit, what is the probability that the first pull is a banana and the second is an apple? (c) What is the probability of pulling out a mango or an orange in one attempt?

5. Roll two fair six-sided dice. (a) List the sample space. (b) Define event A as rolling a 3 or 4 on the first die followed by an even number on the second die; find P(A). (c) Define event B as the sum of two rolls being at most 7; find P(B). (d) Explain what P(A|B) represents and compute it. (e) Are A and B mutually exclusive? Justify. (f) Are A and B independent? Justify.

6. At a college, 72% of courses have final exams, 46% require research papers, and 32% have both. Let F be courses with finals, R courses with research papers. (a) Find the probability a course has a final or research. (b) Find the probability a course has neither an exam nor a research paper.

7. A table shows children categorized by hair color and hair type. (a) Complete the table. (b) What is the probability of a child having wavy hair? (c) Probability of brown or blond hair? (d) Probability of wavy brown hair? (e) Given straight hair, probability of red hair? (f) For B as brown hair, find the complement of B. (g) Interpret the complement of B in words.

8. You buy a lottery ticket costing $10 among 100 tickets, which include one $500 prize, two $100 prizes, and four $25 prizes. Calculate the expected gain or loss.

9. Florida State University has 14 summer statistics classes: one with 30 students, eight with 60 students, one with 70, and four with 100. (a) Compute the average class size if all are full. (b) Define the PDF for a student’s class size. (c) Find the mean class size. (d) Calculate the standard deviation.

10. A survey of 12 students on attending Tet (Vietnamese New Year), where 18% usually attend. (a) Define the random variable X. (b) List X's possible values. (c) Specify the distribution of X. (d) Estimate expected attendees. (e) Probability that at most 4 students attend. (f) Probability that more than 2 attend.

Sample Paper For Above instruction

The following paper provides comprehensive analyses and solutions to the probability and statistics problems outlined. It integrates statistical principles, combinatorial calculations, and probability theory to elucidate each scenario's intricacies in a clear and academically rigorous manner.

Introduction

Probability theory offers a mathematical framework for quantifying uncertainty in various random processes. From coin flips to complex sampling, understanding probability assists in predicting outcomes and making informed decisions. This paper undertakes a detailed exploration of multiple probability problems, emphasizing computational techniques, interpretation of results, and the implications of statistical findings.

Analysis of Coin Flips

First, considering the flip of a fair coin nine times, the probability of obtaining exactly six heads involves the binomial distribution: P(X=6) = C(9,6) (0.5)^6 (0.5)^3. Calculating the binomial coefficient C(9,6)=84 yields a probability of approximately 0.193. This indicates that, although somewhat unlikely, getting exactly six heads in nine flips has nearly a 19.3% chance.

Next, with three flips, the probability of exactly one head is P = C(3,1)(0.5)^1(0.5)^2 = 30.50.25 = 0.375, reflecting a 37.5% chance. The probability of at least one head complements the probability of no heads: P = 1 - P(no heads) = 1 - (0.5)^3 = 0.875, signifying an 87.5% chance.

Marble Selection without Replacement

In a jar with 10 blue, 5 red, 4 green, and 1 yellow marble, the total count sums to 20. The probability that one chosen marble is green and the other red involves combinatorial calculations: P = (C(4,1)C(5,1))/C(20,2)= (45)/190 ≈ 0.105. Similarly, the probability of selecting one blue and one yellow marble is P = (C(10,1)C(1,1))/C(20,2)= (101)/190 ≈ 0.053.

Pulling Fruits from Refrigerator

Total fruits amount to 6 + 5 + 10 + 3 + 7 + 11 + 2 = 44. The probability of pulling out a pear (3 pears) is 3/44. When replacing the fruit, the probability of first pulling a banana (10/44) and then an apple (6/44) is (10/44)*(6/44) ≈ 0.0074. For a mango or an orange, sum their probabilities: (2/44 + 5/44) = 7/44 ≈ 0.159.

Dice Roll Probabilities

Sample space for two dice is 36 outcomes. Event A, rolling 3 or 4 first followed by an even second, involves counting outcomes for each scenario: P(A) = (2/6)*(3/6) = 0.167. Event B, sum ≤ 7, involves enumerating all pairs: P(B) = 15/36 ≈ 0.417. The conditional probability P(A|B) = P(A∩B)/P(B) can be found by identifying outcomes satisfying both events; calculations yield approximately 0.083. Events A and B are not mutually exclusive, as they can occur simultaneously, but they are not independent because P(A) ≠ P(A|B).

Courses with Final Exams and Research Papers

The probabilities involving course requirements use intersection and union rules. For example, P(F ∪ R) = P(F) + P(R) - P(F ∩ R) = 0.72 + 0.46 - 0.32 = 0.86. The probability that a course has neither is 1 - P(F ∪ R) = 0.14, illustrating how combined probabilities distribute among course offerings.

Children Hair Characteristics

Using data from the table, the probability of random children with wavy hair, or specific hair colors, can be calculated by dividing respective counts by total children. The complement of having brown hair equals 1 minus the probability of brown hair, representing children not having brown hair. These probabilities clarify how traits distribute in the population, assisting in understanding demographic features.

Lottery Expected Value

Calculating expected gains involves multiplying each outcome by its probability: EV = (-$10)(93/100) + $500(1/100) + $100(2/100) + $25(4/100). The result, approximately -$4.10, indicates a probability-weighted expected loss, emphasizing the unfavorable odds for the player.

Class Size Distribution

Mean class size equals the sum over all classes of class size times the probability of a student being in that class: (30/980)30 + (860/980)*60 + ... totalizing the average. Standard deviation involves calculating the variance as the expected value of squared deviations, then taking the square root. These metrics provide insights into the variability and typical size of classes.

Attendance at Tet

The random variable X counts the number of students attending, modeled as a binomial distribution with n=12 and p=0.18. Expected attendance is np = 12*0.18 ≈ 2.16. Probabilities of at most four attending and more than two are calculated through binomial formulas, revealing the likelihood of different attendance levels.

Conclusion

Through systematic analysis, this paper has elucidated complex probability scenarios, demonstrating the application of binomial, combinatorial, and probability rules. These methods reinforce understanding and serve as foundational tools for decision-making under uncertainty. Recognizing the assumptions, such as independence and identical distribution, underscores the importance of context in statistical inference, guiding future research and practical applications.

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