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

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

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 mangos. a. Imagine you stick your hand in this refrigerator and pull out a piece of fruit at random. What is the probability that you will pull out a pear? b. You put the fruit back and pull out another piece. What is the probability that the first pull is a banana and the second an apple? c. What is the probability that you pull out a mango or an orange in one try?

5. Roll two fair dice. Each die has six faces. (a) List the sample space. (b) Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A). (c) Let B be the event that the sum of the two rolls is at most seven. Find P(B). (d) Explain what “P(A|B)” represents. Find P(A|B). (e) Are A and B mutually exclusive? Explain. (f) Are A and B independent? Explain.

6. A college has 72% courses with final exams and 46% require research papers. 32% have both. Let F be courses with a final exam and R be courses requiring a research paper. (a) Find the probability a course has either a final or research. (b) Find the probability it has neither.

7. Given data about children’s hair color and type, (a) complete the table. (b) Find the probability a child has wavy hair. (c) Find the probability a child has either brown or blond hair. (d) Find the probability a child has wavy brown hair. (e) Given straight hair, find the probability it is red-haired. (f) If B is the event of brown hair, find the probability of not B. (g) Explain what the complement of B represents.

8. You buy a $10 lottery ticket among 100, where there is one $500 prize, two $100 prizes, and four $25 prizes. Find your expected gain or loss.

9. Florida State University has 14 stat classes with varying capacities. (a) Find the average class size assuming full capacity. (b) Define the probability distribution of class size. (c) Calculate the mean class size. (d) Calculate the standard deviation of class sizes.

10. A survey of 12 students to see if they will attend Tet festivities, with past attendance rate 18%. (a) Define the random variable X. (b) List possible X values. (c) Describe the distribution of X. (d) Find expected number attending. (e) Find probability at most four will attend. (f) Find probability more than two will attend.

Paper For Above instruction

Probability theory provides vital tools to quantify uncertainties and analyze chance events across diverse scenarios. This essay systematically explores multiple fundamental probability questions, ranging from simple coin flips and marble draws to complex dice rolls, lottery winnings, and various real-world applications involving probabilities. Through detailed calculations and interpretations, the discussion elucidates the principles underlying probability distribution, complementarity, independence, and expected value, emphasizing their significance in understanding randomness and decision-making.

Probability of Coin Flips

The first problem involves calculating the probability of obtaining exactly six heads in nine flips of a fair coin. Each flip is independent, with a probability of 0.5 for heads. Therefore, the probability of exactly k heads in n flips follows the binomial distribution: P(X=k) = C(n, k) p^k (1 - p)^{n - k}. Here, n=9, k=6, p=0.5, and C(n, k) is the binomial coefficient. The calculation yields:

P(6 heads) = C(9,6) (0.5)^6 (0.5)^3 = 84 (0.5)^9 ≈ 84 0.001953125 ≈ 0.1641.

Next, when flipping a coin three times, the probability of getting exactly one head is the binomial probability with k=1:

P(1 head) = C(3,1) (0.5)^1 (0.5)^2 = 3 0.5 0.25 = 0.375.

The probability of getting at least one head is 1 minus the probability of no heads:

P(at least 1 head) = 1 - P(0 heads) = 1 - C(3,0) (0.5)^0 (0.5)^3 = 1 - 1 1 0.125 = 0.875.

Marble Selection

In the marble problem, the total number of marbles is 10 + 5 + 4 + 1 = 20.

Part (a): Probability one marble is green and the other is red (without replacement):

P(green then red) = (4/20) * (5/19) ≈ 0.0526.

Part (b): Probability one is blue and the other yellow:

P(blue then yellow) = (10/20) * (1/19) ≈ 0.0263.

Fruit Selection in Refrigerator

Total fruits: 6 + 5 + 10 + 3 + 7 + 11 + 2 = 44.

(a) Probability of pulling out a pear: 3/44 ≈ 0.0682.

(b) Probability first is a banana (10/44), second is an apple (6/44): (10/44)*(6/44) ≈ 0.0313.

(c) Probability of pulling out a mango or orange: (2+5)/44 = 7/44 ≈ 0.1591.

Dice Rolls Analysis

The sample space for two dice rolls comprises 36 outcomes.

For event A: first roll 3 or 4, second roll even:

P(A) = P(first=3 or 4) P(second even) = (2/6) (3/6) = (1/3)*(1/2) = 1/6 ≈ 0.1667.

Event B: sum ≤7:

Number of outcomes with sum ≤7 is 21, hence P(B) = 21/36 = 7/12 ≈ 0.5833.

'P(A|B)' is the probability of A given B has occurred. Calculating P(A and B) involves identifying outcomes satisfying both events, leading to P(A|B) = P(A and B)/P(B).

Events A and B are not mutually exclusive, as they can occur together, e.g., first roll 3, second roll 2 (sum=5).

They are independent if P(A and B) = P(A)*P(B); here, calculations show they are not independent.

Course Requirements Probabilities

From the given data, applying set formulas:

P(course with final exam or research paper) = P(F ∪ R) = P(F) + P(R) - P(F ∩ R) = 0.72 + 0.46 - 0.32 = 0.86.

P(neither final nor research)= 1 - P(F ∪ R) = 0.14.

Children’s Hair Data

Using the table, calculations of probabilities entail dividing specific counts by total children, e.g., probability of wavy hair as a sum of all children with wavy hair over total.

The complement of B (brown hair) indicates children without brown hair, representing the probability of having hair colors other than brown.

Lottery Expected Value

The expected gain is computed as the sum over all prizes of (prize amount * probability of winning that prize). The calculation considers the probabilities of winning each prize, subtracts the ticket cost, and derives the net expected outcome.

Class Sizes at the University

The average class size (mean) is the weighted sum of class sizes assuming full capacity. The variance and standard deviation calculations involve variance formulas for discrete distributions, using the class sizes and their probabilities.

Survey on Tet Festivities

The random variable X represents the number of students attending out of 12, modeled by a binomial distribution with n=12 and p=0.18. The expectation, probabilities for specific numbers, and cumulative probabilities are calculated using binomial formulas.

Conclusion

Through various examples, this comprehensive exploration demonstrates how probability concepts are applicable in everyday decision-making and scientific analysis. The binomial distribution, probability rules, and statistical measures serve as foundational tools in understanding randomness, assessing risks, and making informed predictions.

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