In the following, the capital letters A, B, or C are used to represent some event like “it will rain tomorrow” or”a computer chip is defective”. The notation P(A) then stands for the probability of event A.An experimentis any process yields a result or observation. Experiments do not necessarilyinvolve boiling flasks of strangely colored liquids or electricity arcingbetween shiny metal poles, like you might see in a movie depiction of Dr.Frankenstein”s laboratory; they can be as simple as asking someone what theirfavorite color is.
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The classicalor theoretical definition ofprobability assumes that there are a finite number of outcomes in a situationand all the outcomes are equally likely.
ClassicalDefinition of Probability
Though you probably have not seen this definition before, you probably have an inherent grasp of the concept. In other words, you could guess the probabilities without knowing the definition.
Cardsand Dice The examples that follow require someknowledge of cards and dice. Here are the basic facts needed computeprobabilities concerning cards and dice.
A standard deck of cards has four suites: hearts,clubs, spades, diamonds. Each suite has thirteen cards: ace, 2, 3, 4, 5, 6, 7,8, 9, 10, jack, queen and king. Thus the entire deck has 52 cards total.
When you are asked about the probability of choosinga certain card from a deck of cards, you assume that the cards have been well-shuffled,and that each card in the deck is visible, though face down, so you do not knowwhat the suite or value of the card is.
A pair of diceconsists of two cubes with dots on each side. One of the cubes is called a die, and each die has six sides.Each side of a die has a number of dots (1,2, 3, 4, 5 or 6), and each number of dots appears only once.
Example1 The probability of choosing a heart from adeck of cards is given by
Example 2 Theprobability of choosing a three from a deck of cards is
Example 3 Theprobability of a two coming up after rolling a die (singular for dice) is
The classical definition works well in determiningprobabilities for games of chance like poker or roulette, because the statedassumptions readily apply in these cases. Unfortunately, if you wanted to findthe probability of something like rain tomorrow or of a licensed driver in Louisiana being involvedin an auto accident this year, the classical definition does not apply.Fortunately, there is another definition of probability to apply in thesecases.
EmpiricalDefinition of Probability
The probability of event Ais the number approached by
as the total number of recorded outcomes becomes “verylarge.”
The idea that the fraction inthe previous definition will approach a certain number as the total number ofrecorded outcomes becomes very large is called the Law of Large Numbers. Because of this law, when the ClassicalDefinition applies to an event A,the probabilities found by either definition should be the same. In other words, if you keep rolling a die,the ratio of the total number of twos to the total number of rolls shouldapproach one-sixth. Similarly, if you draw a card, record its number, returnthe card, shuffle the deck, and repeat the process; as the number ofrepetitions increases, the total number of threes over the total number ofrepetitions should approach 1/13 ≈ 0.0769.
In working with the empirical definition, most of the timeyou have to settle for an estimate of the probability involved. This estimateis thus called an empirical estimate.
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Example 4 To estimate the probability of a licensed driver in Louisiana being involved in an auto accident this year, you could use the ratio
To do better than that, you could use the number ofaccidents for the last five years and the total number of Louisiana drivers in the last five years. Orto do even better, use the numbers for the last ten years or, better yet, thelast twenty years.
Example 5Estimating the probability of rain tomorrow would be a little more difficult. Youcould note today”s temperature, barometric pressure, prevailing wind direction,and whether or not there are rain clouds that could be blown into your area bytomorrow. Then you could find all days on record in the past with similartemperatures, pressures, and wind directions, and clouds in the right location. Your rainfall estimate would then be the ratio
To make your estimate better, you might want to add inhumidity, wind speed, or season of the year. Or maybe if there seemed to be norelation between humidity levels and rainfall, you might want add in the daysthat did not meet your humidity level requirements and thus increase the totalnumber of days.
Example 6 If you want to estimate the probability that a dam will burst, or a bridge willcollapse, or a skyscraper will topple, there is usually not much past dataavailable. The next best thing is to do a computer simulation. Simulation results can be compiled a lot faster with a lot less money and less loss of life than actual events. The estimated probability of say a bridge collapsing would be given by the following fraction
The more true to life the simulation is, the better theestimate will be.
Basic ProbabilityRules For either definition, the probability of an event A is always anumber between zero and one, inclusive; i.e.
Sometimes probability values arewritten using percentages, in which case the rule just given is written asfollows
If the event A is not possible, then P(A) = 0 or P(A) = 0%. If event A is certain to occur, then P(A) = 1 or P(A)= 100%.
The sum of the probabilities for each possible outcome ofan experiment is 1 or 100%. This is written mathematically as follows using thecapital Greek letter sigma (S)to denote summation.
Probability Scale* The bestway to find out what the probability of an event means is to compute theprobability of a number of events you are familiar with and consider how theprobabilities you compute correspond to how frequently the events occur. Untilyou have computed a large number of probabilities and developed your own senseof what probabilities mean, you can use the following probability scale as arough starting point. When you gain more experience with probabilities, you maywant to change some terminology or move the boundaries of the differentregions.
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*This is a revised and expanded version of the probability scale presented in Mario Triola, Elementary Statistics Using the Graphing Calculator: For the TI-83/84 Plus, Pearson Education, Inc. 2005, page 135.