Discuss The Following Concepts And Give Examples From Everyd

discuss The Following Concepts And Give Examples From Everyday Life

Discuss the following concepts and give examples from everyday life in which you might encounter each concept. Consider the experiment of arriving for class, where some possible outcomes are not arriving (missing class), arriving on time, and arriving late.

a) Sample space

The sample space in this context includes all possible outcomes of the event "arriving for class": not arriving (missing class), arriving on time, and arriving late. This set encompasses every possible outcome of the experiment. For example, if we consider a typical day where students arrive for class, the sample space might be S = {missing class, arriving on time, arriving late}.

b) Probability assignment to a sample space

Assigning probabilities involves estimating how likely each outcome is within the sample space. There can be more than one valid way to assign these probabilities, depending on the context or data available. For instance, if historical data shows that students arrive on time 70% of the time, arrive late 20% of the time, and miss class 10% of the time, those can be assigned as probabilities to each outcome. Alternatively, if no data is available, one might assign equal probabilities to each outcome, assuming they are equally likely, i.e., each with a probability of 1/3.

Probabilities can be estimated through relative frequencies observed over multiple trials or occurrences. For example, over 100 class days, if students arrived on time 70 days, the estimated probability of arriving on time is 70/100 = 0.7. When events are equally likely, the probability of each outcome can be computed as 1 divided by the total number of outcomes. For instance, with three outcomes in the sample space, each would have a probability of 1/3.

ii. How can probabilities be estimated by relative frequencies?

Probabilities can be estimated by relative frequencies obtained through empirical observations over time. By recording the number of times an event occurs relative to the total number of trials, we can approximate the probability. For example, if out of 200 days, students arrived late 40 days, then the relative frequency of arriving late is 40/200 = 0.2, which serves as an estimate of its probability.

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Discussion of Probability and Statistics

While the course emphasizes statistics, understanding the distinction between probability and statistics is essential as they are interconnected fields but serve different purposes.

Probability is the mathematical study of random phenomena. It involves predicting the likelihood of future events based on known or assumed probabilities. Probability problems often involve scenarios where the outcomes are uncertain but can be modeled mathematically. For example, calculating the probability that a die rolled will show a number greater than 4 involves analyzing the possible outcomes and their likelihoods. In this case, the main goal is to determine the probability based on a known or assumed model of the event.

In contrast, statistics involves collecting, analyzing, interpreting, and presenting data to make inferences or decisions based on observed data. Statistical methods are used to analyze data collected from experiments, surveys, or observations to draw conclusions about larger populations. For example, estimating the average time students arrive for class based on a survey of actual arrival times exemplifies statistical analysis. Statistics are especially useful when the probabilities are unknown, and we seek to understand or infer the characteristics of a population from sample data.

Problems where probability is the main tool include forecasting weather, calculating insurance risks, or determining the chances of winning a game based on known odds (e.g., roulette, dice). These problems involve known models or assumptions about the likelihood of different outcomes, and probability helps predict future events or assess risks.

Problems where statistics is the main tool involve analyzing data collected from real-world observations, such as estimating average incomes in a city, determining the effectiveness of a new drug based on clinical trials, or evaluating customer satisfaction survey results. These problems require summarizing data, identifying patterns, and making inferences under uncertainty where the underlying probabilities are initially unknown or complex.

Expected outcomes or conclusions differ: probability provides theoretical or expected likelihoods before data collection, whereas statistics offers empirical insights after analyzing actual data. Both are crucial for informed decision-making in science, business, and everyday life.

References

• Glen Cowan, Design and Analysis of Experiments, Wiley, 1997.
• John K. Kruschke, Doing Bayesian Data Analysis, CRC Press, 2014.
• William M. Bolstad, Introduction to Bayesian Statistics, Wiley, 2004.
• Richard De Veaux, Paul Velleman, David Bock, Stats: Data and Models, Pearson, 2016.
• Andrew Gelman et al., Bayesian Data Analysis, CRC Press, 2013.
• Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye, Probability & Statistics for Engineering and the Sciences, Pearson, 2012.
• David S. Moore, The Basic Practice of Statistics, W.H. Freeman, 2017.
• Gerald Keller, Probability for Applications, Duxbury, 1987.
• David R. Cox, Principles of Statistical Inference, Cambridge University Press, 2006.
• Baruch Fischhoff, Risk Analysis and Human Error, Springer, 1992.