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A Practical Introduction to Mathematical Probability
What is a probability distribution?
A probability distribution is a mathematical function which
considers all possible outcomes of a process and indicates the
relative likelihood of each particular outcome as a percentage.
Rolling two 6-sided dice has eleven possible outcomes. Its probability
distribution is:
The above probability distribution states that "if you
roll two dice, the probability that the value of the throw will
be 7 is 16.7%". The more times you roll the dice, the more
likely it is that your experimental results will agree with
the distribution's prediction. The smaller the number of experiments,
the more "luck" (or the lack of luck) is a significant
factor. When you keep repeating an experiment, if the probability
is in your favor, you will amass winnings, if the probability
is against you, you will be wiped out.
Summing all of the relative likelihoods of all outcomes of
a probability distribution always equals 100%. This fact is
useful when computing the probability against a specific outcome
or set of outcomes. For example, the probability that a roll
will yield a value other than 7 is 83.3% (100%-16.7%).
When there are many possible outcomes, it is more useful
to deal with outcomes falling into specified ranges. For this,
we use the cumulative distribution function.
What is a cumulative distribution function?
The cumulative distribution function is a way of examining a
probability distribution to answer the question "What is
the probability that an outcome takes a value less than or equal
to x?" The cumulative distribution function for the above
dice example is:
From this cumulative distribution function, we can read that
"the probability of an outcome less than or equal to 9"
is 83%. By doing simple math we can determine that "the
probability of an outcome greater than 6, but less than or equal
to 9" is 41% (83%-42%).
The LogNormal probability distribution
For option analysis, this site uses a probability distribution
known as a lognormal distribution. The special characteristic
about a lognormal distribution is that it models situations
where the probability of geometric increases equal the probability
of geometric decreases.
Examples:
The probability of a stock price doubling (multiply by 2)
equals the probability of a stock going decreasing to half
its original value (divide by 2).
The probability of a $100 stock price increasing to $110 (multiply
by 1.1) is equal to the probability of that stock price decreasing
to $90.90 (divide by 1.1).
The lognormal distribution is the basis of much of that mathematics
used for modeling random movements of stock prices and the
resulting option prices, including the famous Black-Scholes
option pricing equations.
Since we are primarily interested in determining the probability
of prices finishing in specific ranges, we use the cumulative
distribution function of the lognormal probability distribution.
Next Section: The
Role of Probability in Modeling Investments
Previous Section: Evaluating Option Investment Opportunities
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