If you toss a coin 3 times, the possible outcomes are as follows (where H means heads and T means tails): TTT, TTH, THT, HTT, THH, HTH, HHT, HHH.
In total, there are 8 possible outcomes. Of these:
Now, the respective probabilities are 1/8, 3/8, 3/8 and 1/8, as you have just seen. So:
p(0) = p(0 Heads) = 1/8
p(1) = p(1 Head) = 3/8
p(2) = p(2 Heads) = 3/8
p(3) = p(3 Heads) = 1/8
This is a probability distribution; it records probabilities for each possible outcome of the random variable.
Discrete Probability Distribution
A table, graph or rule that associates a probability P(X=xi) with each possible value xi that the discrete random variable X can assume is called a discrete probability distribution. It is a theoretical model for the relative frequency distribution of a population.
A random variable is a quantity whose future outcomes are uncertain. Depending on the characteristics of the random variable, a probability distribution may be either discrete or continuous.
A discrete variable is one that cannot take on all values within the limits of the variable. It can assume only a countable number of possible values. For example, responses to a five-point rating scale can only take on the values 1, 2, 3, 4, and 5. The variable cannot have the value 1.7. The variable "number of correct answers on a 100-point multiple-choice test" is also a discrete variable since it is not possible to get 54.12 problems correct. The number of movies you will see this year, the number of trades a broker will perform next month, and the number of securities in a portfolio are all examples of discrete variables.
A continuous variable is one within the limits of variable ranges for which any value is possible. The number of possible values cannot be counted, and, as you will see later, each individual value has zero probability associated with it. For example, the variable "time to solve an anagram problem" is continuous since it could take 2 minutes or 2.13 minutes, etc., to finish a problem. A variable such as a person's height can take on any value as well. The rate of return on an asset is also a continuous random variable since the exact value of the rate of return depends on the desired number of decimal spaces.
Statistics computed from discrete variables are continuous. The mean on a five-point scale could be 3.117 even though 3.117 is not possible for an individual score.
For any random variable, it is necessary to know two things:
With regard to the range of possible outcomes of a specified random variable: