18
Oct

Probability - the MATHEMATICS OF CHANCE.

Introduction

The concept of probability is fairly easy to grasp. Probability is an indicator or a numerical measure of the LIKELIHOOD that an event will occur. Thus, probabilities could be used as measures of the degree of uncertainty associated with events associated with the game of blackjack for which, as a player, you need to make certain decisions that would make you win, such as:

  • What are the “chances” of getting a 10 or an Ace for my “double down”?
  • What is the “likelihood” of getting a favorable card when hitting my hard 15?
  • How “likely” is it that the dealer’s hole card is a 10 so that buying insurance would be a good option?
  • What are the “odds” in favor of raising my bet to the maximum during the next round?

Probability values are always assigned on a scale from 0 to 1. If the event cannot possibly happen, we say that the probability is equal to zero. If the event is sure or certain to happen, then the probability is equal to 1.  A probability near 0 indicates that the event is UNLIKELY to occur; a probability  near 1 indicates that an event is VERY LIKELY or ALMOST CERTAIN to occur. Other probabilities between 0 and  1 indicates  varying degrees of likelihood that an event will occur. The following figure illustrates this view of probability:

Probability as a Numerical Measure of Likelihood of an Event

Probability can be expressed as a fraction, a concept most everybody is familiar with:

Probability (Event) = P (Event) = N/D

where the numerator, N, represents the ways the event can occur SUCCESSFULLY, and the denominator, D, represents the total number of ways the event can occur both SUCCESSFULLY and UNSUCCESSFULLY. As a fraction, probability  can also be expressed as a decimal. Thus, in Tossing or flipping a coin, we will observe one of two possible results: get a HEAD or TAIL. With a fair coin,  getting a HEAD or getting  a TAIL are equally likely. Hence, the probability of the event “get a HEAD”, is given by

P(HEAD) =  1/2 = 0.50

Experiments and the Sample Space

In discussing probability, we first define a (PROBABILITY) EXPERIMENT as any process that results in well-defined outcomes. When the experiment is repeated, one and only one of the possible experimental outcomes can occur. Several examples of such experiments in the gaming world are found in the following table:

 

In analyzing a particular experiment, it is necessary to carefully  identify the experimental outcomes. The set of all possible experimental outcomes is usually referred to as the SAMPLE SPACE for the experiment; any one of the experimental outcomes is called a SAMPLE POINT and is an element  of the sample space.

It should be mentioned that each outcome of a probability experiment occurs at random.  This means that you cannot predict with certainty which outcome will occur when the experiment is conducted. In addition, each outcome of the experiment is equally likely to occur. This also means that each outcome has the same probability of occurring.

It is also important to point out that the notion  of a probability experiment is somewhat different from the “experiments” conducted in science laboratories. In the laboratory, the researcher assumes that each time an experiment is repeated in exactly the same way, the same outcome will occur. When we speak of probability experiments, the outcome is DETERMINED BY CHANCE, such that even though the experiment might be repeated in exactly the same way, a different outcome may occur.

For this reason, probability experiments may also be referred to as RANDOM EXPERIMENTS.

Events

In discussing probabilities, it is usual to consider several outcomes of the experiment.  In our earlier example of rolling  a single die, we may want to consider getting an odd number -  a 1, 3 or 5. We  refer to this as the event of getting an odd number  from the experiment of rolling a single die. Thus, an event  would consist of one or more outcomes of the sample space.   An event with only one outcome is called a simple event. An event with  two or more events is called a compound event. 

In the experiment of drawing a card from a standard single-deck, we can list the outcomes of  several events, as shown in the following  table:

Assigning Probabilities to Experimental Outcomes

In assigning probabilities to experimental outcomes, two basic requirements / axioms must be satisfied:

  1. The probability values assigned to each experimental outcome (or sample point) must be between 0 and 1.  Denoting by Ei the ith experimental outcome and P(Ei) its corresponding probability, we must have

0 ≤  P(Ei)  ≤ 1  for all  i

  1. The sum of all of the probabilities must be equal to one. That is,

P(E1) + P(E2) +  . . . P(Ek)  = 1

Any method  that satisfies the above requirements and results in reasonable numerical measures of the likelihood of the outcome is acceptable.

In practice, the classical or objective method, the relative frequency method, or the subjective method are often used. This will be discussed in our next post.

 

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