Informal introduction to some properties of probability
Whether probability is defined through sampling from a finite population, or sampling from a hypothetical infinite population, it obeys the same rules.
We only informally introduce some of the ideas here. It is easiest to understand them in the context of sampling a single value from a finite population.
Probabilities are always between 0 and 1
For any event, A,
0 ≤ P(A) ≤ 1
This follows from the fact that probabilities are really proportions.
Meaning of probabilities 0 and 1
For any event, A,
If the event A cannot happen then P(A) = 0
If the event A is certain to happen then P(A) = 1
State of bank account
Consider the event that a savings account has a negative balance. The probability of this event is 0 since savings accounts are not allowed to go into debt.
The probability that a cheque account is 'either in credit or has zero balance or is in debt' is 1 since it is certain that it will be in one of the three states.
Probability that an event does not happen
For any event, A,
P(A does not happen) = 1 - P(A)
Customer purchases
If a proportion 0.2 of customers entering an electrical shop make a purchase, then a proportion (1 - 0.2) = 0.8 do not make a purchase. Since probabilities are really proportions, the same result holds for them.
Addition law
When two events cannot happen together, they are said to be mutually exclusive. For any two mutually exclusive events, A and B,
P(A or B) = P(A) + P(B)
If the events A and B are not mutually exclusive,
P(A or B) < P(A) + P(B)
Frequent flyer flights
Let X denote the number of airline flights taken in a year by a member of the airline's frequent flyer programme. The possible values for X are 0, 1, 2, ..., and these values are mutually exclusive.
If P(X = 0) = 0.1, P(X = 1) = 0.3, P(X = 2) = 0.3, P(X > 3) = 0.3, then the probability that the person will have fewer than 2 flights is P(X = 0) + P(X = 1) = 0.1 + 0.3 = 0.4.
Independence
When sampling two or more values at random with replacement from a population, the choice of each value does not depend on the values previously selected. The successive values are then called independent.
In random sampling with replacement, or random sampling from an infinite population, successive values are independent.
On the other hand, if sampling without replacement from a finite population, successive sample values are not independent. The second value selected cannot be the same as the first value, so knowing the first value affects the probabilities when the second value is selected.
In random sampling without replacement from a finite population, successive values are not independent.
Independence can be given a more precise definition, but this informal definition is enough for our purposes here.
Rolling dice
When two or more dice are rolled, this is usually done such that the second die has probability 1/6 for each value, irrespective of the value that appeared on the first die. The values that appear in the first and second dice are therefore independent.
More about probability
In a later chapter, we will extend some of these ideas about probability.