PROBABILITY
How Probability Quantifies Randomness
Phenomena are any observable occurrences.
- With a small number of observations, outcomes of random phenomena may look quite
different from what you expect.
- With a large number of observations, summary statistics settle down and get increasingly
closer to particular numbers.
As we make more observations, the proportion of times that a particular outcome occurs gets
closer and closer to a certain number we would expect.
This long-run proportion provides the basis for the definition of probability.
With random phenomena, the proportion of times that something happens is highly random and
variable in the short run but very predictable in the long run.
For example, as one tosses a die, in the long run the number of times a 6 occurs becomes more
predictable and less random. It gets closer and closer to 1/6.
We will interpret the probability of an outcome to represent long-run results.
With any random phenomena, the probability of a particular outcome is the proportion of times
that the outcome would occur in a long run of observations.
Since a probability is a proportion, it takes a value between 0 and 1.
Example: a roll of a fair die has outcome 6 with probability 1/6 in a long run of observations.
Different trials of a random phenomenon are independent if the outcome of any one trial is not
affected by the outcome of any other trial.
Example: With independent trials, whether you get a 5 on one roll of a fair die does not affect
whether you get a 5 on the following roll.
The law of large numbers, which gamblers invoke as the law of averages, only guarantees long-
run performance. In the short run, the variability may well exceed what you expect.
We sometimes can find probabilities by making assumptions about the nature of the random
phenomenon. For instance, by symmetry, it may be reasonable to assume that the possible
outcomes of that phenomenon are equally likely.
In theory, we could observe several trials of a random phenomenon and use the proportion of
times an outcome occurs as its probability. In practice, this is imperfect. The sample proportion
merely estimates the actual probability, and only for a very large number of trials is it necessarily
close.
