Transcript From Randomness to Probability

From Randomness to
Probability
Vocabulary







Random Phenomenon: all possible outcomes are known, but it is
not known which values did or will happen
Trial: a single event or realization of a random phenomenon
Outcome: the value measured, observed, or reported for an
individual instance of a trial
Event: a collection of outcomes
Independence: knowing the outcome of one event does not alter
the probability that the other event occurs
Law of Large Numbers: the long-run relative frequency of
repeated independent events gets closer and closer to the true
relative frequency as the number of trials increases
Probability: a number between 0 and 1 that reports the likelihood
of the event’s occurrence
Vocabulary






“Something has to Happen Rule” : the sum of the probabilities of
all possible outcomes of a trial must be 1
Complement Rule: The probability of a event occurring is 1
minus the probability that it doesn’t occur
P(A) = 1 –
P(Ac)
Disjoint: events that share no outcomes in common; also called
“mutually exclusive”
Addition Rule: if A and B are disjoint events, then the probability
of A or B is
P(A U B) = P(A) + P(B)
Legitimate Probability Assignment: an assignment of probabilities
to outcomes is legitimate is each probability is between 0 and 1
and the sum of the probabilities is 1.
Multiplication Rule: If A and B are independent events, then the
probability of A and B is P(A U B) = P(A) x P(B)
Dealing with Random
Phenomena


With random events, the possible outcomes
are known, but it is unsure which outcome
will occur.
Random phenomena are a part of everyday
life. For example, if we drive through an
intersection with a traffic light everyday, the
light has two possible outcomes, red or
green. Although the light may follow a
pattern, it is unknown when you will arrive at
the light, meaning the outcome is random.
Probability



Probability is an event’s long-run relative
frequency.
Probability may not help us predict a
particular outcome, but it can help us see
long-term trends.
For example, to find the probability of
catching a red at the traffic light can’t be
tracked in a day or a week. It should tracked
for a month or longer for accuracy.
The Law of Large Numbers



The LLN says the long-run relative frequency of
repeated independent events gets closer and closer
to the true relative frequency as the number of trials
increases
The law is often misunderstood because the longrun is difficult to grasp. Probabilities even out only in
the long-run. The long-run must be infinitely long to
give them time to even out.
A common misconception occurs in probability. For
example, gamblers often believe that a number that
hasn’t come up on the roulette wheel or in the lottery
is “due” to hit. The problem is that probabilities even
out only in the long run.
Probability (again)

Relative frequencies settle down in the long
run—and the result is probability

Probability MUST fall between 0 and 1
Personal Probability



Probability that can’t be based on long-run
behavior
Personal Probability doesn’t display the
consistency of relative frequency probabilities
The probability of you getting an A in statistics
class is personal probability. We can only
take the class once, so the probability of
getting an A in the class can not be based on
long-run behavior.
Probability MUST be Between
0 and 1


If the Probability is 0, the event never occurs.
If the probability is 1, the event always
occurs. It can’t possibly be greater then 1 or
less then 0.
A probability is a number between 0 and 1.
For any event A, 0≤P(A)≤1.
Something Has to Happen



If a random phenomenon has only one possible
outcome, it’s not very interesting. The probabilities
must be distributed among two or more outcomes.
Something always occurs, making the probability of
something happening 1. This is called the
“Something has to happen rule.”
The probability of the set of all possible outcomes of
a trial must be 1.
P(S) = 1. (S represents the set of all possible
outcomes)
The Complement Rule


The set of outcomes that are not in the event
A is called the complement of A, or AC.
The Complement Rule states that the
probability of an event occurring is 1 minus
the probability that it doesn’t occur.
P(A) = 1 – P(AC)
The Addition Rule


Events that can’t occur together are disjoint,
or mutually exclusive.
For two disjoint events, A and B, the
probability that one or the other occurs is the
sum of the probabilities of the two events.
P(A U B) = P(A) + P(B), provided A and
B are disjoint
The Multiplication Rule


The Multiplication Rule says that for
independent events, to find the probability
that both events occur, we just multiply the
probabilities together.
For two independent events, A and B, the
probability that both A and B occur is the
product of the probabilities of the two events.
P(A ∩ B) = P(A) × P(B), provided that A and
B are independent