Title: Chapter 12: Probability
1Chapter 12 Probability
- Basic Concepts
- Events Involving Not and Or
- Conditional Probability Events Involving And
- Binomial Probability
- Expected Value
2Chapter 12 Probability
- 3. Conditional Probability Events Involving
And
3Conditional Probability
- Define The probability of event B, computed on
the assumption that event A has happened, is
called the conditional probability of B given A.
It is denoted as
4Example Conditional Probability
A red and green die are rolled. Event A is that
the sum of the dice equals 6. Event B is that
the same number is rolled on each die.
Find the conditional probability that the same
number is rolled on each die given that the sum
of the dice equals 6.
5Example Conditional Probability
The sample space is given on page 673 and also on
the next slide.
6Example Conditional Probability
7Example Conditional Probability
- Event Asum of dice is 6
- has cardinality 5
- Event same number is rolled on each die given
that the sum of the dice equals 6 - has cardinality 1.
8Example Conditional Probability
9Conditional Probability
- Notice from the last example that the sample
space is reduced from S with - to
- where
10Conditional Probability Formula
11Previous Example Conditional Probability
- Event Asum of dice is 6
- Event B same number is rolled on each die
- Intersection
12Previous Example Conditional Probability
- Sample space has 36 members
- Therefore,
13Conditional Probability
14Previous Example Conditional Probability
15Independent Events
- Define Two events A and B are called independent
events if the occurrence of one of them has no
effect on the occurrence of the other one.
16Independent Events
- Define Two events A and B are independent events
means - or
17Independent Events
- Example of independent events
- Roll a pair of dice and the events are Aeven on
the first and Bodd on the second
18Independent Events
- 2. Example of events that are not independent
- Roll a pair of dice and the events are Aeven on
the first and Bsum of dice is a four
19Multiplication Rule of Probability (Events
Involving And)
- If A and B are any two events, then
20Proof Multiplication Rule of Probability (Events
Involving And)
- Multiply both sides by P(A)
- to get the formula
21Example Multiplication Rule of Probability
(Events Involving And)
- Two marbles are drawn without replacement from a
bag containing two green, three yellow, and four
red marbles. Find the probability that you will
draw - Two green marbles
- No yellow marbles
22Example Multiplication Rule of Probability
(Events Involving And)
- Answers
- Two green marbles 1/36
- No yellow marbles 5/12
23Multiplication Rule of Probability (Events
Involving And)
- If A and B are any two independent events, then
24Proof Multiplication Rule of Probability (Events
Involving And)
- Independent events means that
- to get
25Example Multiplication Rule of Probability
(Events Involving And)
- Two marbles are drawn with replacement from a bag
containing two green, three yellow, and four red
marbles. Find the probability that you will
draw - Two green marbles
- No yellow marbles
26Example Multiplication Rule of Probability
(Events Involving And)
- Answers
- Two green marbles 4/81
- No yellow marbles 4/9
27Example Multiplication Rule of Probability
(Events Involving And)
- Not Independent Page 753, problem 26 (see page
698 for card problems) - Independent Page 754, problem 41,42,43 (note
that 43 is the complement of problem 42!) - Not Independent Page 754, problem 46
- Independent Page 755, problems 71,72,73
28Example Genetics
- Genetics of Plant Height
- Consider a population of plants that cannot
self-pollinate - The plant is diploid and has two copies of each
gene one from the ovule (egg) and one from the
pollen. - Height is determined by a single gene which has
two alleles (variants) - b and B
29Example Genetics
- The B allele is dominant
- Plants with at least one copy of B are tall and
those with two copies of b are short. - Suppose that the parent plants are both with
genotype Bb (one short allele and one tall
allele)
30Example Genetics
- What is the probability that the next generation
will be short?
31Example Genetics
- Answer
- Sample space of genotypes for next generation
- (see example on page 729)
32Example Genetics
- Answer
- Aevent next generation is short
- Probability next generation is short
-
33Example Genetics
- Suppose we are given a tall offspring, but we do
not know its genotype. What is the probability
that its genotype is BB?
34Example Genetics
- Answer
- This is an example of conditional probability
- Tevent that plant is tall
- Gevent that genotype is BB
- Find
35Example Genetics
- Answer
- Tevent that plant is tall
- Gevent that genotype is BB
36Example Genetics
37Example Colorblindness
- The probability of colorblindness depends on a
persons sex. - Mevent a person is male
- Fevent that person is female
- Cevent that person is colorblind
- Assume the following
38Example Colorblindness
- Find the probability that a male is colorblind.
- also
- Find the probability that a colorblind person is
a male.
39Example Colorblindness
- Answer
- Find the probability that a male is colorblind.
- 5 of all males are colorblind
40Example Colorblindness
- Answer
- Find the probability that a colorblind person is
a male. - Need to determine
41Example Colorblindness
- Answer
- The event that you are colorblind
- Colorblind and Male OR Colorblind and Female
- Note
- and
- are mutually exclusive (do not intersect)
42Example Colorblindness
43Example Colorblindness
- Answer
- Find the probability that a colorblind person is
a male. - 83 of colorblind people are male