PROBABILITY & STATISTICS
A. Define the Law of Large Numbers, citing a credible source.
A "law of large numbers" is one of several theorems expressing the idea that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero. (1)
Basically, this law means that with many, many repeated trials of a given event, the results will approach the average, expected mean of the population. Eventually the actual average will stabilize around the expected average of occurrences.
This phenomenon can be readily observed amongst the major stock market exchange services within the United States. Although the stock market may fluxuate daily, weekly or even yearly, when observed over the life of the U.S. stock market, the market has returned an approximate average of 10%. That is the reason most reputable stock market advisors suggest that investors “buy and hold” stock forever. (2)
The Law of Large Numbers can be divided into two categories. The Weak Law states that the sample average converges in probability towards the expected value. The Strong Law states that the sample average converges closely to the expected value.
People who gamble at chance games in Los Vegas, though, make the mistake of thinking that the games or cards will magically “remember” that after so many bets, the gamblers number will eventually “hit” or occur. This is far from the truth. In order to increase his or her chances of sustaining a consistently high average of wins, the gambler would have to continue to bet hundreds of times. This is impractical even for the wealthiest gambler!
B. Explain the Law of Large Numbers in your own words using a coin toss as an example.
A classic example is the tossing of a coin. The theoretical probability of landing on either heads or tails is 50-50%. But in reality, subsequent tosses may actually result in a 40-60% or a 60-40% outcome....