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Perceptions of risk
Published in Martin Loosemore, John Raftery, Charles Reilly, David Higgon, Risk Management in Projects, 2012
Martin Loosemore, John Raftery, Charlie Reilly, Dave Higgon
To repeat, the law of large numbers states that the mean calculated from a sample will become closer to the underlying population mean as the sample gets larger. Means calculated from small samples are less stable than means based on large samples. Logically, therefore, the correct answer is B. Namely, the smaller sub-contractor will be more likely to submit the greater number of variant readings, where its projects are deviating more than 10 per cent over time. This disadvantages the small sub-contractor. However, over a large number of experiments, psychologists have found that people tend to return an answer of A or C by ignoring the law of large numbers.
Ensemble Methods The Wisdom of the Crowd
Published in Chong Ho Alex Yu, Data Mining and Exploration, 2022
If the logic of bagging is conceptualized by the law of large numbers, the connection between data science and traditional statistics is even more obvious. In probability and statistics, the law of large numbers guarantees that as the sample size increases, the sample statistics can get closer to the population parameter (Seneta 2013). A typical example is that a casino may lose to the gambler in a single game, but in the long run the average outcomes eventually favor the casino. In a similar vein, a single weak learner might be defeated by the noisy data set, but repeated attempts lead to the final victory for the strong learner.
Advanced Concepts
Published in Kim H. Pries, Jon M. Quigley, Testing Complex and Embedded Systems, 2018
The law of large numbers suggests that we increase our confidence when we increase the sample size as long as the samples are truly random samples. We need to be on the alert for heteroscedasticity to avoid issues with clumping of data. Most simple confidence calculations have an inbuilt assumption of homoscedasticity that can be challenged.
Convergence behavior of statistical uncertainty in probability table for cross section in unresolved resonance region
Published in Journal of Nuclear Science and Technology, 2023
CLT is a well-known method for quantifying statistical uncertainty [9]. In accordance with the law of large numbers, the mean of the random samples drawn from a population tends to approach the mean of the population when the sample size is sufficiently large. This implies that the mean of the random samples will tend to a normal distribution when the sample variance is finite. This low is applicable regardless of the normality of the original population. CLT uses this property to calculate the standard error. CLT is widely used to quantify statistical uncertainty since it works even if the original population does not follow a normal distribution. CLT is an effective method, however, it requires a large number of samples. The standard error may not be accurate when there are not enough samples.
Assessing the effect of repair delays on a repairable system
Published in Journal of Quality Technology, 2020
Jiaxiang Cai, Candemir Cigsar, Zhi-Sheng Ye
When m is large, we recommend the probabilistic quasi-Monte Carlo method (Morokoff and Caflisch 1995). When the integration region is a unit cube, a conventional Monte Carlo method samples N points from the region, computes the corresponding values of the integrand, and uses the average to approximate the integral. The law of large numbers ensures the convergence of the average to the true value. The quasi-Monte Carlo accelerates the convergence through choosing sampling points by design. The resulting quasi-random sample points are more evenly distributed than traditional ones, allowing for a smaller approximation error. In many practical situations, it converges with an error that diminishes nearly as rapidly as independent of the dimension m. With a change of integration variables the integration region in model [3] becomes a unit hypercube. By generating N quasi-random samples from the unit hypercube, the integral can be approximated. Our simulation experience suggests that the adaptive subdivision is slightly more accurate, but the quasi-Monte Carlo is much faster when m is large, say, m > 10.
Larger versus Luckier: preservice teachers’ exploration of probabilistic reasoning through an aleatoric music activity
Published in International Journal of Mathematical Education in Science and Technology, 2023
Song An, Alyse Hachey, Daniel Tillman, Danielle Divis, Bryn Birdwell
Ten of the preservice teachers exhibited confidence when unexpected outcomes happened, and they retained their original theoretical prediction even when their observed results were dissimilar to what they assumed would occur at the onset of chance music making. These participants showed a deep understanding of randomness during chance processes and a higher tolerance for relying on theoretical probability in the face of (seemingly) contradictory evidence. However, while they continued to rely on correct theoretical probabilistic thinking, the skewed nature of their data did seem to test their confidence and produced a related emotional response. For instance, Alexandria documented how her emotions changed during the chance music composition activity: Based on my experiment I noticed it was harder to [get] La (THT) to come up in the coin toss. Out of 48 tosses, our group only got 2 La in the music. I did not worry about the patterns during the first part of the experiment. On part two, I was a little stressed that we didn’t get all the musical notes. Due to not having at least four in La in music, we had to repeat the toss 14 more times to be able to obtain 4 of each musical notes. Mathematically speaking, there was no need to stress. I think this is completely random, it was just ‘luck’. I withhold the same perspective, which is that it’s very probable that each musical notes can be obtained since there is 50% chance that each side could be gotten despite I have La repeats much less frequently in this music composition. How funny is that? It was a bit confusing for me at first, but at the end it was hilarious to see the awkward results. If I continue to do more tossing, with a greater number of times, I am sure I will eventually get the same amount of each musical note in music.The law of large numbers (Gal, 2005) is the leading theory for combatting the issue of ‘luck’ in chance processes; this law asserts that as the number of trials increases the experimental probability will be approaching the theoretical probability. From her writing, it seems apparent that Alexandria has a robust understanding of the Law of Large Numbers and this knowledge held strong, despite the cognitive and emotional conflicts she described feeling during the chance music composing experience.