Page 4.1 (C:\data\StatPrimer\probability.wpd Print date: 8/1/06)
What is probability?
The probability of an event is its relative frequency (proportion) in the population.
An event that happens half the time (such as a head showing up on the flip of a fair coin) has probability
50%. A horse that wins 1 in 4 races has a 25% probability of winning.
A treatment that works in 4 of 5
patients has an 80% probability of success.
Probability can occasionally be derived
by counting the number of ways a thing can happen and
determine its relative frequency. To use a familiar example, there are 52 cards in a deck, 4 of which are
Kings. Therefore, the probability of randomly drawing a King = 4 ÷ 52 = .0769.
Probability can be
. If an event occurs
times out of
, then its probability
becomes large. For example, if we flip a coin many times, we expect to see
half the flips turn up heads. This experience is unreliable when
but becomes increasingly
increases. For example, if a coin is flipped 10 times, there is no guarantee that we will
observe exactly 5 heads. However, if the coin is flipped 1000 times, chances are better that the
proportions of heads will be close to 0.50.
Probability can be used to quantify
opinion. If a doctor says “you have a 50% chance of
recovery,” the doctor
that half of similar cases will recover in the long run. Presumably, this is
based on knowledge, and not on a whim. The benefit of stating subjective probabilities is that they can be
tested and modified according to experience.
Range of possible probabilities:
Probabilities can be no less than 0% and no more than 100% (of
Let A represent an event. Then, Pr(A) represents the probability of the event.
represent the complement of event A. The complement of an event is its
“opposite,” i.e., the event
happening. For example, if event A is recovery following treatment,
represents failure to recover.
Law of complements:
) = 1
). For example, if Pr(
) = 0.75, then Pr(
) = 1
0.75 = 0.25.
A random variable is a quantity that varies depending on chance. There are two
types of random variables,
Discrete random variables
can take on a finite number of possible outcomes. We study
binomial random variables as examples of discrete random variables.
Continuous random variables
form an unbroken chain of possible outcomes, and can take on
an infinite number of possibilities. We study Normal random variables as examples of
continuous random variables.