Probability is about certain things (“events”) occurring, and how likely they are to occur in a special way.
There is always a context: a prescribed kind of events. For example, suppose there are several different balls in a bag, and a few are pulled out. One kind of event would be the particular set of balls being pulled out. Another kind would be sequences of balls being pulled out one at a time, keeping track of the order in which they are pulled out.
The basic idea of probability is to compare the number of possible events of this certain kind, with the number of a special subset of those events.
The probability of the special events is the ratio of the number of special events to the total number of events within the context.
So probability is based on the counting of events.
A probability is a number between 0 and 1. Sometimes probabilities are stated in terms of percentages, going from 0 to 100%, or in terms of odds, usually expressed as a ratio (for example, 1:1000).
Consider a bag containing 10 different balls, and the case of pulling out 5 of the balls. What is the probability of getting a specific sequence of 5 balls? The set of events is that of all permutations of 5 balls taken from 10, and the special event is the single case of the specific sequence.
In this case, there is just 1 of the specific sequence of 5 balls, and there are P(10, 5) possible permutations of 10 balls taken 5 at a time. The probability is then
The probability that a certain set A of events will occur is written symbolically as
Sets of events can be combined in several ways. Two important ways are: events of one set or another, and events of one set and another.
The rest of this page describes formulas for the probabilities of such combinations of sets of events, and important special cases.
The joint probability of two sets of events A and B is the probability that an event of either one or the other set occurs. That is, that an event of the set (A or B) will occur.
The two sets of events A and B are assumed to be from the same context, so the total number of ways for all things in that context to happen is the same for both sets. So their probabilities, as ratios, have the same denominators.
The number of ways an event could be in set A or set B is often simply the sum of the number of ways it can be in A and the number of ways it can be in B. However, it may happen that an event could be in both sets. In case an event is in both A and B, you don’t want to count its occurrence twice, so the probability of one occurrence of each event shared by both sets has to be subtracted away from the sum.
Therefore, the probability of an event of set A or set B occurring is generally
That is,
When two events can not both happen, they are said to be mutually exclusive.
In case A and B are mutually exclusive, pr( A and B ) = 0, so the formula for their joint probability is simplified:
For instance, in one roll of a die, it cannot happen that the dice lands with both 2 and 5 up. The results of rolls of a die are considered mutually exclusive.
(What is the probability of throwing a 1 or 2 or 3 or 4 or 5 or 6 on a single die?)
On the other hand, when drawing cards, drawing a heart is not mutually exclusive of drawing a queen, because the queen of hearts is both a heart and a queen.
The conditional probability is the probability of one event occurring given that another event occurs.
For example, the probability that a child has blue eyes given that just one parent has blue eyes.
The probability of A occurring given that B occurs is written with the special notation
In discussions of conditional probability, there is always at least a tacit sense of knowledge that something happens, usually stated in terms of order: the probability of A given that B happened.
Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other.
Successive tosses of a coin or a die are considered to be independent.
An important example of non-independence is the case of drawing cards from a deck of cards without returning the drawn cards to the deck. The probability of drawing a certain card depends on what cards were drawn previously.
The formula for conditional probability is:
If A is independent of B, then
In case of independent events the above formula is simplified:
A symmetric way of writing conditional probabilities is called Bayes’ rule: