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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.

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

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