Basic Probabilities

Probability is a measure of the likelihood of an event. It is frequently calculated as a rational number, but often converted to a percentage.  Rational numbers are numbers which can be expressed as a ratio of two whole numbers.  So if we say the probability, P, of something happening is 1 chance in 2, we can express this in several ways:

\[ P=\frac{1}{2}=0.5=50\% \]

The above example happens to relate well to a simple real world example, the coin flip.  In a single coin flip, we will either get a heads or a tails.  So, if we ask the question "What is the probability of getting a heads in a coin flip?" we can answer "One chance in two, or 50%.

An interesting feature of probabilities of cumulative events is that they are not additive.  To illustrate this, we need the cumulative probability equation.  The cumulative probability, \( P_C \), is a function of multiple individual probabilities, \( P_1, P_2...P_n \), with the relationship,

\[ P_C=1-(1-P_1) \times (1-P_2) \times ...(1-P_n). \]

So if we ask the question, "What are the chances of getting heads at least once during two coin flips?" we cannot say 100% for a very important reason.  If we get a tails on the first try and try again, the coin does not know that it was tails last time, so the chances of getting a heads on the second coin flip are still 50%.  To get the answer to this question, we must use the cumulative probability equation.  The cumulative probability, {tex inline}P_C{/tex}, of getting a heads on either of two coin flips, \( P_1 \) and \( P_2 \), is

\[ P_C = 1-[(1-P_1)\times(1-P_2)] = 1-[(1-0.5)\times(1-0.5)] = 0.75 = 75\%. \]

Notation in Probability

As you can see from the examples above, probability is often noted with a P.  More formally, the probability of event A occurring is written as

\[ P(A). \]

The probability of events A and B occurring is written as

\[ P(A \cap B). \]

The probability of events A or B occurring is written as 

\[ P(A \cup B). \]

Finally, we need some notation for a relation of an event to a condition.  Conditions are also a form of events, so this can be a bit confusing.  The probability of event A occurring, given that event B has occurred, is noted as

\[ P(A|B). \]

As an example of how we would write a conditional probability, lets make a statement based on the data in Koester's book. In a Search and Rescue operation, the probability, P, of locating a subject in an injured condition, I, given that the subject is a solo male, M, is 9%. We would state this as follows: The probability of an injury, given that the subject of a search is male, is 9%, or

\[ P(I|M) = 9\%. \]

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