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Chapter 14
From Randomness to Probability
Dealing with Random Phenomena
 A random phenomenon is a situation in which we know what
outcomes could happen, but we don’t know which particular
outcome did or will happen.
Probability
 The probability of an event is its long-run relative
frequency.
 While we may not be able to predict a particular individual
outcome, we can talk about what will happen in the long run.
 Each attempt of a random phenomenon is called a trial
 Example: each time you flip a coin
 Each trial will generate an outcome which are the
individual possibilities
 Example: each number we see on top when we roll a die.
Probability continued
 The set of all possible outcomes of an event is the sample space
of the event.
 Example: Rolling a dice
 S = {1, 2, 3, 4, 5, 6}
 An event is an outcome (or a set of outcomes) from a sample
space. It is usually denoted by a capital letter. The probability
of event A is denoted as P(A)
 Example 1: When flipping three coins, an event may be getting the result
THT In this case, the event is one outcome from the sample space.
 Example 2: When flipping three coins, an event may be getting two tails. In
this case, the event is a set of outcomes HTT, THT, TTH from the sample
space.
Law of Large Numbers
 The Law of Large Numbers (LLN) says that the long-run
relative frequency of repeated independent events gets closer
and closer to the true relative frequency as the number of
trials increases.
 For example, consider flipping a fair coin many, many times.
The overall percentage of heads should settle down to about
50% as the number of outcomes increases.
The Law of Large Numbers (cont.)
 The common (mis)understanding of the LLN is that random
phenomena are supposed to compensate some for whatever
happened in the past. This is just not true.
 For example, when flipping a fair coin, if heads comes up on
each of the first 10 flips, what do you think the chance is that
tails will come up on the next flip?
 Thanks to the LLN, we know that relative frequencies settle
down in the long run, so we can officially give the name
probability to that value.
Slide 14- 6
Formal Rules of Probability
1)
The probability of any event is between 0 and 1.
0  P( A)  1
A probability of 0 indicates that the event will never occur and a
probability of 1 indicates that the event will always occur.
Formal Probability (cont.)
2.
“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.)
Slide 14- 8
Formal Probability (cont.)
3.
Complement Rule:
 The set of outcomes that are not in the event A is called the

complement of A, denoted AC.
The probability of an event occurring is 1 minus the
probability that it doesn’t occur: P(A) = 1 – P(AC)
Example: When flipping two coins, the probability of getting
two heads is 0.25 and the probability of not getting two heads is
0.75
Slide 14- 9
Formal probability (cont.)
4) Events A and B are said to be disjoint if they have no
outcomes in common. The probability that one or the other
occurs is the sum of the probabilities of the two events.
P( A or B)  P( A)  P ( B )
P( A  B)  P( A)  P( B)
Formal rules (cont.)
4) Disjoint example:
Let event A be rolling a die and landing on an even number and
event B be rolling a die and landing on an odd number.
The outcomes for A are _____________ and the outcomes
for B are ______________. These two events are disjoint
because they have no outcomes in common. What is the
probability?
Formal rules (cont.)
5) The outcome of one trial does not influence or change
the outcome of another trial it is said to be independent.
Example: When you flip a coin, it does not affect whether the next
flip will be heads of tails.
If events A and B are independent then the probability of A and
B equals the probability of A times the probability of B.
P( A and B )  P ( A)  P ( B )
P( A  B)  P( A)  P( B)
Formal rules (continued)
5) Independent example
The probability that the yellow die lands on an even number
and the red die lands on an odd number is: