Suppose an event has a number of possible outcomes, and the theoretical probability of each outcome is known. This collection of data is called the probability distribution of the event. We can make a histogram of the distribution; in each case, the height of a column represents the probability that the value of the event lies in the range corresponding to the column. For example, suppose three quarters are tossed,  and the number of heads is recorded. For convenience, suppose the three quarters are labeled 1, 2 and 3. There are eight possible throws: HHH, HHT, HTH, HTT,  THH, THT, TTH, TTT, where the three letters represent the result (Head or Tail)  of the toss of coins 1, 2 and 3 in that order. Each of these throws is equally likely,  with probability 1/8, and exactly one must occur. So the probability of 0 heads is 1/8 (TTT must have occurred); one head has probability 3/8 (HTT, THT or TTH);  two and three heads have probabilities 3/8 and 1/8. The histogram is

Sample Problem1.1 Supposed a biased coin has a 40% probability of landing heads on any throw. In an experiment, the coin is tossed three times and the number of heads is recorded. Find the probability distribution of the result and draw the histogram for this probability distribution.

Solution. The probability of rolling the sequence HHH is 0.4×0.4×0.4= .064;  for HHT it is 0.6 × 0.4 × 0.4 = .096; and so on. The following table shows the probabilities, and the sum of probabilities for each number of heads:

Suppose each outcome of the event has a numerical value. Then the mean of a probability equals the expected value. If there are n possible values, x₁,x₂,...,x_n,  and the probability of value xi is pi, then the mean is Σⁿ_i=1 p_ix_i. The mean is often denoted by µ, the Greek letter “mu” that corresponds to m.

The median is the value such that the probability of this result or a smaller one is 50%. The first and third quartiles, q₁ and q₃, are the 25% and 75% values respectively; for example, the probability of a result smaller than or equal to the first quartile is 25%. The interquartile range is the difference q₃ − q₁. All of these quantities are analogous to the corresponding ones for samples.

In many cases, the median and quartiles are most easily found by calculating the cumulative probability of the event. The cumulative probability cp(x) of a value x is the probability that the outcome will be equal to or less than that value. That is,  it is the sum of the probabilities of all outcomes with value less than or equal to x.

The cumulative probability distribution of the event is defined analogously to the probability distribution, using cumulative probabilities instead of probabilities. The first quartile is the smallest value x with cp(x) ≥ 0.25; the median and third quartile are defined similarly, with 0.5 and 0.75 replacing 0.25.

The standard deviation of a probability distribution is defined analogously to the standard deviation of a set of data. Using the same notation as we did for the mean,  the squared deviation corresponding to value xi is (x_i − µ)², and the mean squared deviation, or variance, is Σⁿ_i=1 p_i(x_i −µ)². The standard deviation equals the square root of the variance. It is commonly denoted σ (the lower-case Greek letter “sigma,”  which corresponds to s).