An Experiment Has Three Steps With Three Possible Outcomes

An Experiment Has Three Steps With Three Outcomes Possible For The

Analyze various probability and statistics problems involving experiments, sampling, conditional probability, independence, and joint probabilities. The tasks include calculating the total number of experimental outcomes, constructing tree diagrams, listing outcomes, computing probabilities for different scenarios, understanding the concepts of mutually exclusive and independent events, applying Bayes' theorem, and interpreting survey data related to social media usage and television show ratings.

Sample Paper For Above instruction

Introduction

Probability theory and statistical analysis are fundamental tools in understanding the likelihood of events and making inferences about populations. This paper addresses multiple problems involving these concepts, demonstrating the application of basic probability rules, combinatorial calculations, conditional probabilities, the distinction between independence and mutual exclusivity, and the interpretation of survey data. Each problem is examined in detail, providing insights into the problem-solving strategies employed in statistics and probability.

Problem 1: Experiment with Multiple Steps and Outcomes

An experiment consists of three steps, with different possible outcomes at each step: three outcomes for the first step, two outcomes for the second, and four outcomes for the third. The question asks for the total number of possible experimental outcomes for the entire experiment. Applying the fundamental counting principle, the total outcomes are calculated as the product of outcomes at each step:

Total outcomes = 3 (first step) × 2 (second step) × 4 (third step) = 24.

This straightforward approach exemplifies the basic principle of counting in probability.

Problem 2: Tossing a Coin Twice

The experiment involves tossing a coin two times, with 'H' denoting heads and 'T' tails. To visualize all possible outcomes, a tree diagram can be constructed, starting from the first toss and branching into two outcomes ('H' and 'T'), then further branching for the second toss. The sample space includes all ordered pairs of outcomes:

• (H, H)
• (H, T)
• (T, H)
• (T, T)

The probability of each outcome, assuming a fair coin, is 1/4, since each individual outcome is equally likely. If outcomes are not equally likely, probabilities would need to be assigned accordingly based on the specific biases of the coin.

Problem 3: Number of Samples in Random Selection

Given a population of 50 bank accounts, selecting a sample of 4 accounts without replacement involves calculating the number of combinations:

Number of possible samples = C(50, 4) = (50 × 49 × 48 × 47) / (4 × 3 × 2 × 1) = 230,300.

This reflects the total different subsets of size four that can be chosen from the population, emphasizing the combinatorial nature of sampling without replacement.

Problem 4: Survey Data on Global Warming and Age Groups

The survey data categorize responses (Yes, No, Unsure) by age groups (18-29, 30+). Calculations involve finding probabilities of specific responses within age groups:

• Probability that a respondent aged 18-29 thinks global warming will not pose a serious threat: calculated using the proportion of 'No' responses in that age group.
• Similarly, for respondents aged 30+.
• The overall probability of answering 'Yes' is based on the combined responses across all respondents.

Differences between age groups can be identified by comparing these probabilities, shedding light on age-related perceptions of climate change risks.

Problem 5: Car Rental Behavior

From survey data: 45.2% rented a car for business, 55% for personal, and 20% for both. Probabilities of union and complement events are computed:

• Probability of renting for at least one reason (business or personal): P(B ∪ P) = P(B) + P(P) - P(B ∩ P) = 0.452 + 0.55 - 0.20 = 0.802.
• Probability of not renting for either reason: 1 - P(B ∪ P) = 0.198.

This illustrates the application of set theory and probabilities in understanding overlapping events.

Problem 6: Conditional and Joint Probabilities

With P(A) = 0.50, P(B) = 0.50, and P(A ∩ B) = 0.20, conditional probabilities are calculated as:

• P(A | B) = P(A ∩ B) / P(B) = 0.20 / 0.50 = 0.40.
• P(B | A) = P(A ∩ B) / P(A) = 0.20 / 0.50 = 0.40.

Since P(A | B) = P(A), events A and B are not independent—as independence would require P(A | B) to equal P(A).

Problem 7: Mutually Exclusive vs. Independent Events

For P(A) = 0.3 and P(B) = 0.7, with P(A ∩ B) = 0 (mutually exclusive), P(A | B) = P(A ∩ B) / P(B) = 0 / 0.7 = 0. The probability P(A | B) ≠ P(A), indicating that mutually exclusive events are dependent, not independent. The statement that mutually exclusive events are independent is false, because the occurrence of one prevents the other, contradicting independence.

Problem 8: Bayes' Theorem and Conditional Probability

Given prior probabilities P(A₁) = 0.50, P(A₂) = 0.50, and P(A₁ ∩ A₂) = 0, and conditional probabilities P(B | A₁) = 0.10, P(B | A₂) = 0.06, the calculations involve:

• Checking mutual exclusivity: P(A₁ ∩ A₂) = 0, confirming mutual exclusivity.
• Calculating joint probabilities: P(A₁ ∩ B) = P(B | A₁) × P(A₁) = 0.10 × 0.50 = 0.05.
• P(A₂ ∩ B) = 0.06 × 0.50 = 0.03.
• P(B) = P(A₁ ∩ B) + P(A₂ ∩ B) = 0.05 + 0.03 = 0.08.
• Applying Bayes' theorem:

1. P(A₁ | B) = P(A₁ ∩ B) / P(B) = 0.05 / 0.08 = 0.625.
2. P(A₂ | B) = 0.03 / 0.08 = 0.375.

This demonstrates how Bayesian updating modifies prior probabilities based on new evidence.

Problem 9: Social Media Usage and Voice Opinions

The joint probability table is constructed from survey data, considering the number of females and males who use social media. For example, if total respondents are 1,331, the probability that a respondent is female and uses social media is calculated as:

P(Female ∩ Uses Social Media) = (Number of females using social media) / 1,331.

The conditional probability that a respondent uses social media given they are female is obtained by dividing the joint probability by the probability of being female. Testing for independence involves checking if P(Female ∩ Uses) = P(Female) × P(Uses).

The analysis reveals whether gender and social media usage are independent variables based on the data.

Problem 10: Television Show Ratings

The survey data on viewer ratings are used to compute probabilities. The probability that a viewer rates the show as average or better is:

P(Average or Better) = (Number rating average + number rating above average + number rating excellent) / Total respondents = (12 + 15 + 12) / 51 ≈ 0.84.

The probability of rating below average or worse (poor or below average) is:

P(Poor or Below Average) = (4 + 8) / 51 ≈ 0.23.

These calculations help to understand viewer perceptions and ratings distribution, informing producers and marketers.

Conclusion

The problems addressed illustrate the practical application of probability principles in varied contexts, from simple experiments to survey data analysis. They highlight the importance of understanding the distinctions between dependent and independent events, the use of combinatorics in sampling, and the application of Bayesian inference. Proper interpretation of survey data enables researchers to draw meaningful conclusions about public opinion and behavior. Overall, mastery of these concepts is essential for rigorous statistical analysis and data-driven decision making.

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