Think of any experiment with chance outcomes – buying a lottery ticket, betting on a horse race, going on a blind date, undergoing some medical treatment. We use the word distribution to specify all the possible outcomes, along with their associated probabilities. (We slipped in that word when writing about Poisson’s analysis of how many rare events will happen, given a large number of opportunities.)
The ‘distribution’ is central to analysing the range of consequences from a chance experiment. Plainly, we need to be clear about the full extent of the possible outcomes. To give sensible values for their probabilities, we must spell out our assumptions, and hope that they are appropriate for the experiment we seek to investigate.
Discrete distributions
First, we look at circumstances where the possible outcomes can be written as a list, each outcome having its own probability. The phrase discrete distribution applies here.
The most straightforward case is when we can count the number of outcomes, and agree that they should all be taken as equally likely. The term uniform distribution is used, as the total probability is spread uniformly over the outcomes. Many experiments are expected to fit this bill – roulette, dice, hands of cards, selecting the winning numbers in a lottery, etc. Accurate counting generates the appropriate answer.
Recall the term ‘Bernoulli trials’ to mean a sequence of independent experiments with a constant probability of Success each time. With a fixed number of Bernoulli trials, there is a simple formula, called the binomial distribution , that gives the respective probabilities of exactly 0, 1, 2, . . . Successes. This formula depends only on the number of trials, and the Success probability. As you run through the outcomes in order, their probabilities initially increase up to a maximum value, then fall away towards zero. (Poisson distributions also follow this pattern.)
We expect a binomial distribution for the number of Sixes among twenty throws of a die; or the number of correct answers when a student guesses randomly among five choices at each of thirty questions on a multiple choice exam. But we do not expect it when asking how many Clubs a bridge player has among his thirteen cards: although each separate card has probability one quarter of being a Club, successive cards are not independent, as the chance of a Club on the next card is affected by all previous outcomes.