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What does probability matter?

Robert Hayes Published: September 21, 2013

Ever wonder what probability is, maybe why some poker hands are worth more than others or perhaps even what the odds are on some other random outcome?  The simplest and least descript definition for probability is a ratio of counts.  The numerator of the ratio is the number of ways you can obtain a success and the denominator of that ratio is the total number of possible outcomes.  Take for example the rolling of a single die, this gives a one sixth probability of getting any given single number.  Here the successful outcome is the desired number from the die and the total number of possible outcomes are all six values from a standard die so that the ratio ends up being one divided by six.

When counting the total number of possible outcomes to apply this method, these values when taken from two independent events must be multiplied.  For example when rolling a pair of dice, there are 6 independent outcomes from each die but when counting how many combinations for the pair, this number is then 36=6×6.  When you have a success that can only be obtained in one way, the probability for this success (such as a pair of two’s) is then one divided by thirty six.  Again, this is just the total number of ways of getting a success divided by the total number of possible outcomes.

If a success can be attained in more than one way, then the total number of ways a success can be obtained have to be summed up for the numerator in the ratio to calculate the total probability.  Getting the pair of dice to sum up to three for example can be done in only two ways (with each die being either a one and a two respectively or vice versa).  The probability of getting a pair of dice to sum to three is then two divided by thirty six or one eighteenth.

Although this ratio of counts approach works well for small countable systems such as with playing cards and dice, a more complete definition of probability comes from measurement.  If you wanted to know the probability that somebody in Wal-Mart was a democrat, you would have to interview everybody in Wal-Mart to find out what the proportion of people there were of that political affiliation.  Then you would know what the probability is of picking a democrat at random from those at Wal-Mart during the interview.  Once people started coming and going after the thorough interview was over, the probability would change but might still be a good estimate for having that same proportion at any other time.  In principle, the proportion could be biased based on the time of day or even day of the week in that republicans may be more likely not to shop during Sunday or democrats might be more likely to be shopping late at night or vice versa.  Probability can then be an estimate when incomplete knowledge is present or if complete knowledge on a system is available, it becomes simply a counted or measured value.

Public opinion polls are good examples of estimates in that they interview a small portion of a population and assume the same averages apply to the larger population as a whole.  The very way questions are worded or even how subjects are chosen for the poll can bias the result.  As an example, many people might think that asking someone whether they are pro life or anti abortion may be the same question, phrasing it that way could result in different answers from some people.  Counting the ages of students in a classroom or their gender can be a known exact proportion which can then be used to predict the probability of randomly selecting a person from that class who would be a given age or gender.  Even just glancing down the street you can make estimates on how busy traffic is, the average temperature or even wind speed.  All of these being estimates with uncertainty which may or may not be important.

When making scientific measurements of any kind, all these same principles can come to bear.  Sometimes a system may have a probability distribution that can be calculated such as in quantum mechanics but even then, predictions are only useful to the extent that they can be backed up by quality measurements.  Technically, probability and statistics find their way into all measurements and observations but we often don’t realize it.