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The Math Learning Portfolio Casino: Test Your Luck

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Probability Laws
Four Probability Properties
Pascal's Wager and Bernoulli Trials
John Forbes Nash and Game Theory
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Laws of Probability

Laws of Probability

Law of Very Large Numbers

Given enough opportunity, bizarre or odd patterns/results will happen purely based on chance alone. In other words, “luck” is a group phenomenon, where each opportunity provides a chance for a certain type of result to occur. For example, in a lottery, each ticket that is bought adds to the chance of a lucky winner, and a multitude of losers.

 

Law of Nonoccurrence

Some things are so unlikely that there just is not enough opportunity for them to occur. For example, the chance of landing 100 heads or 100 tails in 100 coin tosses is incredibly small. Another example would be if you shook a container with a layer of salt over a layer of pepper, so that they were completely mixed, and then try to restore them to their distinct layers again by simply shaking.

 

Law of Absurdity

Everything is impossible but something must happen. This idea may be best thought through predicting lengths. For example, when one is predicting the length of a trout to be in a certain range, one can never find the exact length, since all measurements are approximations, however the trout is some length.

 

Law of Averages

In numerous trials of an activity, the fraction of occurrences of an event will get closer to the probability of the occurrence in one trial. The method requires an infinite number of trails, but in more practical circumstances, chance variation (the acceptance of small changes within finite repetitions) is necessary. It can be quantified through standard deviation (SD), a formula that provides the range above, and the range below of the predicted outcome in which actual results are likely to sit. This formula appears as follows, when ‘p’ is the chance of occurrence in one trial, where ‘1-p’ equals the chance of nonoccurrence and ‘N’ is the number of trials:

 

SD= √[N x p(1 – p)]

 

 Using this information, one can induce a central limit theorem on the observations, which means one can create probabilities that the results will land in a certain SD ranges. There is a 68% probability that results will be in 1 SD range, 95% in 2 SD ranges, and 99.7% change in 3 SD ranges. Usually if results are above 3 SD ranges, then the results are most likely due to another alternative rather than pure chance and one should reject chance in favour of the alternative.

standarddeviation.gif
Here is a graph showing SD ranges


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