Question

Many, many people have made incorrect assumptions based on probability. Some of the most obvious cases come from people who gamble. For instance, gamblers often believe that if certain events (numbers on a pair of dice or a roulette wheel, or results on a slot machine) have not occurred in some time, then these numbers or results are "due" to occur. Or, they assume that if a repeated behavior results in several losses in a row, then they are "due" for a win if they repeat the same behavior. (Note that most casino games are designed so that the odds are slightly in favor of the casino. The games are otherwise "fair," in terms of their setups and theoretical probabilities for possible outcomes.) Apply the concept of expected values, as well as your understanding of "fair" games, to explain why gamblers are wrong when they use this line of reasoning. Your post must have a minimum of 5 sentences.

Answer 1

The gamblers are wrong when they use the line of reasoning. that if certain events (numbers on a pair of dice or a roulette wheel, or results on a slot machine) have not occurred in some time, then these numbers or results are "due" to occur. Or, they assume that if a repeated behavior results in several losses in a row, then they are "due" for a win if they repeat the same behavior because of the following reason:

By law of large numbers, the probability any 1 specific number, say number 3, out of the 6 numbers of a die is 1/6. So, if a die is tosses 600 times, the number of times the number 3 will occur nearly 600/6 = 100 times.

But, the gambler generally does not make the guess after such large number of trials. For example consider a gambler who does not see the particular number 3 consecutively for 20 trials is almost sure that the number 3 will appear the 21st attempt by so called "law of averages". This observation on the part of the gambler is wrong because number of trials = 20 is still not large to justify the application of law of large numbers.