Example

Example: Consider again the game of Craps. The player rolls a pair of die. If on the first roll of the dice she throws a number other than 2, 3, 7, 11, or 12, the number she throws is the point. The rules say she must roll the dice again and continue to roll until she throws the point and wins, or a 7, and loses. Say the point is 4. Based on the probability model for a pair of dice, on any given roll following the first:

P(win) = P(x = 4) = 3/36.
P(lose) = P(x = 7) = 6/36.
P(roll again) = 1 – P(win) – P(lose) = 27/36 = 3/4.

For each roll, the game either terminates with probability 1/4, or the player must roll again with probability 3/4.

We model this situation with the Bernoulli distribution. Take the two outcomes as “roll again” and “terminate”, and arbitrarily assign the value 0 to the roll again outcome and the value 1 to the terminate outcome. For the example

p= P(terminate) = P(1) = 1/4 and
1-p = P(roll again) = P(0) = 3/4.

We use these probabilities in the discussion for the Geometric Distribution.

Although the Bernoulli is a very simple distribution it is the one most often seen and used by the general public. Many games have individual plays with uncertain results characterized by two alternatives. For example a card randomly drawn from a standard poker deck is an ace or not. The probability that it is an ace is 4/52 (or 1/13). This is the number of aces in the deck divided by the number of cards. The probability that it is not an ace is 12/13. The probability that a coin flip comes up "heads" is 1/2, while the probability of "tails" is also 1/2. These probabilities are determined by the logic of the game and by counting possibilities. If the game is played fairly, the probability is exact. The person who knows the relevant probabilities is a better player than a person who does not.

Some Bernoulli probabilities cannot be determined by logic and counting, but can be estimated through statistics. For example, by polling the voters one can estimate the probability that a specific candidate will win. Similarly, the probability that some automotive part will fail under specified conditions might be estimated by testing a number of the parts and counting the number of parts that fail. Actuaries compute the probabilities of death as a function of age with death and population statistics. These estimates may not be exact, but they are useful in many contexts.

For some uncertain situations, neither logic and counting nor experimentation are possible. Many persons are willing, and sometimes eager, to estimate probabilities and take action. Life includes many gambles that have no sure results. Some make life difficult, but many make life interesting.