Download 7-1

Math 310
Section 7.1
Probability
What is a probability?
Def.
In common usage, the word "probability" is used
to mean the chance that a particular event (or set
of events) will occur expressed on a linear scale
from 0 (impossibility) to 1 (certainty), also
expressed as a percentage between 0 and 100%
http://mathworld.wolfram.com/Probability.html
Sample Space
Def.
The sample space is all the possible results of the
scenario being considered.
Ex.
If I flip a penny, and we assume that it cannot land
on its edge, then there are two possible results: it
lands on heads, or it lands on tails. Thus the
sample space consists of the outcomes heads and
tails.
Notation: S = {H, T}
Ex.
Consider the spinner to the right.
What is the sample space (what
are the possible results)?
D
S = {A, B, C, D}
B
A
C
Outcomes and Events
Def.
The individual results in a sample space are called
outcomes. Events are any set of outcomes (any
subset of the sample space).
Ex.
Again consider the spinner to the right.
What are the outcomes? What
would be an example of an event?
D
The outcomes are: A, B, C and D.
One event would be:
E = {A, B} (read “A or B”)
B
A
C
Experimental probability
Def.
Experimental probabilities are ratios, expressed as
fractions, decimals, or percents, determined by
considering results or outcomes of experiments.
An experiment is an activity whose results can
be observed and recorded.
Theoretical probability
Def.
Theoretical probabilities are ratios, expressed as
fractions, decimals, or percents, determined by
considering results or outcomes under ideal
circumstances.
Exerimental vs. Theoretical
Def.
Consider the toss of a penny. Under ideal
conditions, the penny is not pre-disposed to land
on either side, (ie the penny is equally likely to
land on either heads or tails). We would say the
probability of landing on heads is ½ as is the
probability of landing on tails. However, in
experiments rarely do you get that exact
proportion. (text pg 434)
Bernoulli’s Theorem
Thrm. Law of Large Numbers
If an experiment is repeated a large number of
times, the experimental probability of a particular
outcome approaches a fixed number as the
number of repetitions increase. This number is
the theoretical probability.
Actually determining probabilities
Equally likely events
Def.
When one outcome is just as likely as another the
outcomes are equally likely.
Probability of equally likely events
Thrm.
For a sample space S, with equally likely outcomes,
the probability of an event A is given by:
P(A) = Number of elements of A
Number of elements of S
= n(a)/n(S)
Ex.
Consider a coin. The sample space is S = {H, T}.
Since the outcome of a head is equally likely as a
tail then:
P(H) = ½
P(T) = ½
Ex.
Consider the spinner to the right.
The sample space is
S = {A, B, C, D}. What are the
probabilities of the outcomes,
A, B, C and D?
P(A) = ¼
P(B) = ¼
P(C) = ¼
P(D) = ¼
B
A
D
C
Ex.
Consider the spinner to the right.
The sample space is
S = {A, B, C, D}. What is the
probability of the event
A = {B, C}. (note: to use thrm
the outcomes must be equally
likely.)
P(A) = 2/4 = ½
B
A
D
C
Not all outcomes are always equally likely however.
Ex.
Consider the spinner to the right.
What is the sample space?
What is the probability of the
outcome A?
S = {A, B}
P(A) ≠ ½
A
B
P(A)
Property
The probability of an event is equal to the sum of
the probabilities of the disjoint outcomes
making up the event.
(note: this is useful for figuring out the probability
of an event comprised on outcomes that are not
equally likely.)
Ex.
Consider a die. What is the sample space. What
are the probabilities of the individual outcomes?
What is the probability of rolling an even
number?
S = {1, 2, 3, 4, 5, 6}
P(1) = 1/6, etc.
P(E) = P({2, 4, 6}) = P(2) + P(4) + P(6)
Ex.
Consider the spinner to the right.
What is the sample space?
What is the probability of the
event {A, B}?
A
B
S = {A, B, C}
P({A, B}) = P(A) + P(B) = ½ + ¼ = ¾
C
Mutually exclusive events
Property
If events A and B are mutually exclusive, then
P(A or B) = P(A U B) = P(A) + P(B)
Ex.
Consider a die. What is the probability of rolling
an even number or a 1?
A = {2, 4, 6}, B = {1}
P(A or B) = P(A) + P(B) = ½ + 1/6 = 4/6
Non-Mutually Exclusive Events
If A and B are events the
P(A or B) = P(A) + P(B) – P(A ∩ B)
What is P(A ∩ B)
P(A ∩ B) is the probability of the intersection of
two events.
Ex.
Consider a die. What is the probability of rolling
an odd number or a number divisible by 3?
A = {1, 3, 5}, B = {3, 6}, A∩B = {3}
So P(A or B) = ½ + 1/3 – 1/6 = 4/6 = 2/3
Complementary Events
If A is an event and Ā is its compliment, then
P(A) + P(Ā) = 1,
P(Ā) = 1 – P(A)
Ex.
Consider a die. What is the probability of not
rolling a 6?
A = {not 6} = {1, 2, 3, 4, 5}
So P(A) = 5/6
Question
Why is it that for any event A, 0 ≤ P(A) ≤ 1.
Certain and Impossible Events
A certain event is one whose probability is 1, while
an impossible event is one whose probability is
0.
Notation:
P(Ø) = 0 (impossible event)
P(S) = 1 (certain event)
Properties of Probability
1. P(Ø) = 0 (impossible event)
2. P(S) = 1
3. For any event A, 0 ≤ P(A) ≤ 1.
4. If A and B are events and A ∩ B = Ø, then
P(A or B) = P(A) + P(B).
5. If A and B are events, then
P(A or B) = P(A) + P(B) – P(A∩B)
6. If A is an event, then P(Ā) = 1- P(A)