Download Chapter 4: Probability – The Study of Randomness

Chapter 4: Probability – The Study of Randomness
I.
Randomness (IPS section 4.1 pages 282-287)
A. Random – We call a phenomenon random if individual outcomes are uncertain but
there is nonetheless a regular distribution of outcomes in a large number of
repetitions. Random does NOT mean haphazard. It is a description of a kind of
order that emerges only in the long run.
B. Probability – The probability of any outcome of a random phenomenon is the
proportion of times the outcome would occur in a very long series of independent
repetitions. That is, probability is long-term relative frequency.
II.
Probability Models (IPS section 4.2 pages 287-305)
A. Sample Space – The sample space S of a random phenomenon is the set of all
possible outcomes.
B. Event – An event is an outcome or a set of outcomes of a random phenomenon.
That is, an event is a subset of the sample space.
C. Probability Rules:
1. Any probability is a number between 0 and 1. 0  P(A)  1
2. All possible outcomes together must have probability 1. If S is the sample
space in a probability model, then P(S) = 1.
3. The probability that an event does not occur is 1 minus the probability that the
event does occur. The complement of any event A is the event that A does not
occur, written as Ac. The complement rule states that P(Ac) = 1 – P(A)
4. If two events have no outcomes in common, the probability that one or the
other occurs is the sum of their individual probabilities. Two events A and B
are disjoint if they have no outcomes in common and so can never occur
simultaneously. If A and B are disjoint,
P(A or B) = P(A) + P(B)
This is the addition rule for disjoint events. Disjoint events are also called
mutually exclusive events.
5. Independence and the Multiplication Rule – Two events A and B are
independent if knowing that one occurs does not change the probability that
the other occurs. If A and B are independent, then
P(A and B) = P(A) * P(B)
This is the multiplication rule for independent events. If A and B are NOT
independent, then P(A and B)  P(A)P(B).
NOTE: disjoint events can NOT be independent since the probability of them both
happening is zero, P(A and B) = 0 for disjoint events (p. 296).
D. Venn Diagram – A picture that shows the sample space S as a rectangular area
and events as areas within S.
Figure 4.2 Venn diagram showing disjoint events A and B
Figure 4.3 Venn diagram showing the complement Ac of an event A
Figure 4.4 Venn diagram showing the event {A and B}. This event consists of outcomes common to A and B
E. Probabilities in a Finite Sample Space – Assign a probability to each individual
outcome. These probabilities must be numbers between 0 and 1 must have sum
1. The probability of any event is the sum of the probabilities of the outcome
making up the event.
F. Equally Likely Outcomes – If a random phenomenon has k possible outcomes, all
equally likely, then each individual outcome has probability 1/k. The probability of
any event A is :
P(A) = count of outcomes in A count of outcomes in A
count of outcomes in S
k
Note: a random variable (see below) that has equally likely outcomes has a
uniform distribution.
III.
Random Variables (IPS section 4.3 pages 305 – 318)
A. Random Variable – a random variable is a variable whose value is a numerical.
B. Discrete Random Variable – A discrete random variable has a finite number of
possible values. The probability distribution of X lists the values and their
probabilities:
x1
x2 x3
… xk
Value of X
p1
p2 p3
… pk
Probability
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1.
2. p1 + p2 +…+ pk = 1.
Find the probability of any event by adding the probabilities p i of the particular
values xi that make up the event (as in IIE above).
C. Continuous Random Variable – A continuous random variable X takes all values in
an interval of numbers.
i. Probability Distribution - The probability distribution of X is described by a
density curve. The probability of any event is the area under the density
curve and above the values of Z that make up the event. All continuous
probability distributions assign probability 0 to every individual outcome.
Figure 4.10 The probability distribution of a continuous random variable assigns probabilities under a density curve.
ii. Density Curve – describes the probability distribution of a continuous
random variable. The probability of any event is the area under the curve
above the values that make up the event. Since the ‘area’ of a line is zero,
the probability that a continuous random variable exactly equals any value
is always zero.
iii. Normal Distributions – Normal distributions are one type of continuous
probability distribution. They are bell-shaped, centered at the mean, and
spread about 3 standard deviations in both directions from the mean.
D. Probability Histogram – You can picture a probability distribution by drawing a
probability histogram in the discrete case or by graphing the density curve in the
continuous case.
Figure 4.11 Probability in Example 4.18 as area under a normal density curve.
IV.
Means and Variances of Random Variables (IPS section 4.4 pages 318 – 340)
A. Mean of a probability distribution – the mean of a probability distribution μ, the
Greek letter mu. The mean μ is the balancing point of the probability histogram or
density curve. It is only the center if the distribution is symmetric.
B. Law of Large Numbers – Draw independent observations at random from any
population with finite mean μ. Decide how accurately you would like to estimate μ.
As the number of observations drawn increases, the mean x bar of the observed
values eventually approaches the mean μ of the population as closely as you
specified and then stays that close.
C. Variance and Standard Deviation – The variance σ2 is the average squared
deviation of the values of the variable from their mean. The standard deviation σ
is the square root of the variance. It measures the variability of the distribution
about the mean. It is easiest to interpret from normal distributions.
D. Shift and Scale changes hold for means and standard deviations of random
variables.
Ex: If the mean height of women is 64.5 inches and the standard deviation is 2.5in,
then the mean height and standard deviation of women in centimeters would be
cm = 2.5(cm/in)*in = 161.25cm and cm = 2.5(cm/in)*in = 6.25cm. Remember, shift
changes only affect means and not standard deviations.