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Chapter 6
    Sec 6.1

Chance is all around us.  Probability is the branch of mathematics that descries the pattern of chance outcomes.  The reasoning of statistical inference rests on asking, "How often would this method give a correct answer if I used it very many times?"  When we produce data by random sampling or randomized comparative experiments the laws of probability any that question.  The fundamental concepts of probability are the basis for inference.  The tools you acquire will help you describe the behavior of statistics from random samples and randomized comparative experiments later.

The mathematics of probability begins with the observed fact that some phenomena are random... that is, the relative frequencies of their outcomes seem to settle down to fixed values in the long run.  Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.  You cannot predict the outcome of tossing a coin one time but if you make many, many tosses a regular pattern will emerge....this is the REMARKABLE fact that forms the basis  for probability.    The proportion of tosses that produce heads is quite variable at first, but settles down as we make more and more tosses as shown in Ex. 6.1 on page 331.

Random in statistics is NOT a synonym for "haphazard" but a description of a kind of ORDER that emerges only in the long run.  The idea of probability is empirical which means based on observation rather than theorizing.  When presented with opportunity, some mathematicians tested the coin toss results. 

A little bit of trivia...Count Buffon tossed a coin 4040 times resulting in 2048 heads or 2048/4040 = .5069.  Karl Pearson tossed a coin 24,000 times resulting in 12,012 heads or 12,012/24,000 = .5005.  while a POW during WWII John Kerrich tossed a coin 10,1000 times with 5067 heads or 5067/10,000 = .5067.  These results are extraordinarily alike indicating the conclusion that the probability of tossing a head on a fair coin settles down to around 50%.

A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.  The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Probability is the branch of mathematics that describes random behavior.  Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials.  These trials must be INDEPENDENT, that is, the outcome of one trial must NOT influence the outcome of any other. 

Some statistical history/evolution...Probability theory originated in the study of games of chance.  Tossing dice, dealing shuffled cards, and spinning a roulette wheel are examples of deliberate randomization that are similar to random sampling.  During the 17th century French gamblers asked mathematicians, Blaise Pascal and Pierre de Fermat for help.  Gambling is still with us in casinos and state lotteries.  In the 20th century the mathematics of probability is used to describe the flow of traffic through a highway system, a telephone interchange or a computer processor; the genetic makeup of individuals or populations, the energy states of subatomic particles; the spread of epidemics or rumors, and the rate of return on risky investments.  Although we are interested in probability because of its usefulness in statistics, the mathematics of chance is important in many fields of study.

Let's do Ex. 6.3 on page 334 to practice these basic concepts.

Index