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4: Probability
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
logically
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
estimated
through
experience
. If an event occurs
x
times out of
n
, then its probability
will
converge
on
X
÷
n
as
n
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
n
is small,
but becomes increasingly
reliable as
n
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
subjective
opinion. If a doctor says “you have a 50% chance of
recovery,” the doctor
believes
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.
Notes:
•
Range of possible probabilities:
Probabilities can be no less than 0% and no more than 100% (of
course).
•N
otation:
Let A represent an event. Then, Pr(A) represents the probability of the event.
•
Complement:
Let
}
represent the complement of event A. The complement of an event is its
“opposite,” i.e., the event
not
happening. For example, if event A is recovery following treatment,
then
}
represents failure to recover.
•
Law of complements:
Pr(
}
) = 1

Pr(
A
). For example, if Pr(
A
) = 0.75, then Pr(
}
) = 1
!
0.75 = 0.25.
•
Random variable:
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.
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(4.1)
Binomial Distributions
Binomial Random Variables
Consider a random event that can take on only one of two possible outcomes. Each event is a
Bernoulli
trial
. Arbitrarily, define one outcome a “success” and the other a “failure.” Now, take a series of
n
independent Bernoulli trials. The random number of successes in n Bernoulli trials is
a
binomial
random variable.
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This note was uploaded on 01/28/2010 for the course CMSY 103 taught by Professor N/a during the Fall '06 term at Howard County Community College.
 Fall '06
 n/a

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