Expected or predicted probabilities can be calculated empirically or from functions that predict or model the probability of an outcome or range of outcomes. For example, suppose the result of 100 coin flips is 46 heads and 54 tails. Then the empirical predicted probability of heads is 0.46. The (theoretical) discrete probability distribution of coin flips is 0.5 heads and 0.5 tails.
Instructions
Empirically
1. Find or build a dataset of historical data. For example, counts of the different sums coming up in a craps game may be calculated.
2. Group the events into categories large enough that the probabilities are credible. For example, if the dice weren't thrown enough times that each sum came up several times, then maybe just the two categories (win/lose) may be better, to get reliable predicted probabilities.
3. Divide the counts per category by the total counts. These ratios serve as empirical probabilities per category.
By Discrete P.D.F.
4. Presume that a certain function describes the probability of various outcomes. For example, you may have identified the binomial distribution as predictive of the system of interest. For example, supposed a loaded coin turns up heads four times out of 10. The binomial probability distribution function (p.d.f.) for 10-flip samples says the probability of three heads is (10-choose-3)---(.4)^3(.6)^7.
5. Calculate any relevant permutations. For example, in the above example, 10-choose-3 counts all the combinations of three heads being distributed among 10 positions in the sequence of 10 flips. It is calculated as follows: 10!/[7!3!] = 10---9---8/[3!] = 10---9---8/[3---2] = 120.
6. Finish the calculation and sum all the probabilities for the range of outcomes of interest. Continuing the above example, 120---(.4)^3(.6)^7=0.215. So that's the probability that ten flips of a loaded coin gives exactly three heads. Suppose you want to the probability of getting at least three heads. Then you have to perform the same sort of calculations as above for four heads, for five heads, etc., then sum it all up. Or you can find the probabilities for zero, one and two heads, and add that to 0.215. That will be the probability of getting at most three heads. So just subtract that sum from one to get the probability of getting more than three heads. This latter approach is much faster.
By Continuous P.D.F.
7. Determine what formula applies to the system whose output you are predicting. For example, suppose the normal distribution describes your system well. By the central limit theorem, the normal distribution is a good approximation when a given outcome is a function of several independent variables and for averages of large sample sizes.
8. Perform an integration on the p.d.f. Using calculus, integrate between the values of interest, to determine the predicted probability of outcomes between the two. Note the difference from discrete distributions, where each value of outcome had its own predicted probability.
9. Convert the sample data into standard form to look up in a table, if Step 2 is not a feasible integration. The normal distribution isn't integrated easily by pencil and paper, since a squared term is in the exponent. Tables are used. Data must be converted to standard form first. For the normal distribution, this means subtracting the population mean and dividing by the population standard deviation.
10. Look up the two points in the distribution's table. For example, if after conversion the points of interest for the normal distribution are 0.5 and 1.0, the corresponding cumulative probabilities (starting from the center) are .1915 and .3413. So the predicted probability for outcomes in this range is the difference of the two: 0.1498.
A normal distribution table can be found at the MathIsFun website (see Resources section).
Tags: normal distribution, three heads, predicted probability, probability getting, above example, coin flips, coin flips heads
