Announcements
Finite Probability
Friday, October 14th
I
MyMathLab 5 is due Monday Oct 17
I
Problem Set 5 is due Wednesday Oct 19
Today: Sec. 6.4: Conditional Probability II
Understand the meaning of conditional probability and
independence
Calculate conditional probabilities using the definition,
including equally likely outcomes
Next Class: Sec. 6.5: Tree Diagrams
Cherveny
Oct 14
Math 1004: Probability
Last class: Conditional Probability
Definition
The conditional probability “event E given event F ” is defined as
P(E |F ) =
Cherveny
Oct 14
P(E ∩ F )
P(F )
Math 1004: Probability
Shanghai Population
Example
Twenty percent of the world’s population live in China. The
residents of Shanghai constitute 1.6% of China’s population. If a
person is selected at random from the entire world, what is the
probability that he or she lives in Shanghai?
Answer:
P(Shanghai) =?
P(China) = .2
P(Shanghai|China) = .016
From definition of conditional probability,
.016 =
P(Shanghai)
P(Shanghai ∩ China)
=
.2
.2
So that P(Shanghai) = .016 · .2 = .0032.
Cherveny
Oct 14
Math 1004: Probability
Basketball
Example
Suppose that your basketball team is behind by two points with a
few seconds left in the game. You can try a two-point shot
(probability of success is .48) or a three-point shot (probability of
success is .29). Your shot will be taken just before the buzzer
sounds and each team has the same chance of winning in overtime.
Which shot gives your team the greater probability of winning the
game?
Answer: The three-point shot gives your team a 29% chance of
winning. The two-point shot gives your team a 24% chance of
winning. We multiplied .48 and .5... what lets us do that?
Cherveny
Oct 14
Math 1004: Probability
Independent Events
Definition
Events E and F are called independent if
P(E ∩ F ) = P(E ) · P(F )
Remark: We always know P(E ∩ F ) = P(E |F ) · P(F ), so E and F
independent means that P(E |F ) = P(E ).
Intuitively, E and F are independent means knowledge that F
happens doesn’t influence the likelihood of E .
Cherveny
Oct 14
Math 1004: Probability
Marbles
Example
A sample of two marbles is drawn from a bag containing two white
marbles and three red marbles. Are the events “the sample
contains at least one white marble” and “the sample contains
marbles of both colors” independent?
Answer:
P(at least one white) = 1 −
P(one of each color) =
C (3, 2)
7
=
C (5, 2)
10
C (3, 1) · C (2, 1)
3
=
C (5, 2)
5
P(at least one white ∩ one each color) =
They are not independent since
Cherveny
Oct 14
7
10
·
3
5
6= 35 .
Math 1004: Probability
3
5
Practice
1. If E and F are events with P(E ) = 0.4, P(F ) = 0.5, and
P(E ∪ F ) = 0.7, are E and F independent?
2. Let E and F be independent events with P(E ) = 0.5 and
P(F ) = 0.6. Find P(E ∪ F ).
3. An exam has 10 true-false questions. Are “get first question
right” and “get second question right” independent events?
Why/what assumptions would we be making?
4. If roll a die twice, are the events “the sum is 10” and “a 4 was
rolled” independent?
5. Toss a fair coin five times. What is the probability that heads
appears on every toss, given that heads appears on the first
four tosses?
6. If E and F are independent, show that E 0 and F 0 are also
independent.
Cherveny
Oct 14
Math 1004: Probability
Practice Answers
1. P(E ∩ F ) = .4 + .5 − .7 = .2. Check P(E ∩ F ) = P(E )P(F )
holds, so the events are independent.
2. P(E ∩ F ) = (.5)(.6) = .3. Then P(E ∪ F ) = .5 + .6 − .3 = .8
3. If you guess randomly, then yes they’re independent. But
getting a question right could be evidence that the person
studied, which would influence the probability of other
questions.
3
, P(a 4 is rolled) =
4. Find P(sum = 10) = 36
2
P(sum = 10 and 4 was rolled) = 36
. Since
they are not independent.
5. 1/4 since the flips are independent.
6. Next slide
Cherveny
Oct 14
Math 1004: Probability
11
36 , and
2
11
36 6= 36
·
3
36 ,
#6 Answer
P(E 0 )P(F 0 ) = (1 − P(E ))(1 − P(F ))
= 1 − P(E ) − P(F ) + P(E )P(F )
= 1 − P(E ) − P(F ) + P(E ∩ F )
= 1 − [P(E ) + P(F ) − P(E ∩ F )]
= 1 − P(E ∪ F )
= 1 − P((E 0 ∩ F 0 )0 )
= P(E 0 ∩ F 0 )
We used probability of a complement, definition of independence,
inclusion-exclusion, de Morgan’s Law, and probability of a
complement again!
Cherveny
Oct 14
Math 1004: Probability