Download Math 1004: Probability

Announcements
Finite Probability
Wednesday, October 12th
I
MyMathLab 5 is due Monday Oct 17
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Problem Set 5 is due Wednesday Oct 19
Today: Sec. 6.4: Conditional Probability
Explain the intuitive meaning of conditional probability
Calculate conditional probabilities using the definition,
including equally likely outcomes
Next Class: Sec. 6.4: Conditional Probability II
Cherveny
Oct 12
Math 1004: Probability
Rolling a Die
Example
Roll a fair die. If you know that the outcome is an odd number,
what is the probability that it is a 3?
Answer: There are three odd outcomes, {1, 3, 5}. They are
equally likely. Since we are told the roll was one of them, the
probability of a 3 given the knowledge that the roll is odd is 31 .
Cherveny
Oct 12
Math 1004: Probability
Conditional Probability
Definition
The conditional probability of “event E given event F ” is
P(E |F ) =
P(E ∩ F )
P(F )
Formally: When event F is known to have happened, we think of F
as a new sample space. New events are subsets of F , which are old
events (subsets of the old sample space) intersected with F . And
we divide by P(F ) so that the new sample space has probability 1.
Cherveny
Oct 12
Math 1004: Probability
Rolling a Die
Example
Roll a fair die. If you know that the outcome is an odd number,
what is the probability that it is a 3?
Answer:
P(3|odd) =
P(3 ∩ odd)
=
P(odd)
Outcome
1
3
5
Cherveny
Oct 12
1
6
1
2
=
1
3
Probability
1/3
1/3
1/3
Math 1004: Probability
Conditional Probability Example
Example
A pair of fair dice is rolled. What is the probability that the sum of
the dice is 8, given that exactly one of the dice shows a 3?
Answer:
P(sum is 8 and exactly one 3)
P(exactly one 3)
2/36
1
=
=
10/36
5
P(sum is 8|exactly one 3) =
Note: When outcomes are equally likely we still solve by counting:
P(E |F ) =
Cherveny
# outcomes in both E and F
# outcomes in F
Oct 12
Math 1004: Probability
Math Club
Example
The math club has six sophomore and five freshmen members. If
three members are selected at random for a competition, what is
the probability they are all sophomores, given that at least one is a
sophomore?
Answer:
# teams all sophomore
# teams ≥ 1 sophomore
C (6, 3)
=
C (11, 3) − C (5, 3)
P(all sophomore| ≥ 1 sophomore) =
Cherveny
Oct 12
Math 1004: Probability
Practice
1. Let E and F be events in sample space S. Suppose
P(E ) = 1/2, P(F ) = 1/2, and P(E ∪ F ) = 7/12. Calculate
(a) P(E ∩ F )
(b) P(E |F )
(c) P(F |E )
2. Two cards are drawn one after another from a standard 52
card deck without replacement. What is the probability..
(a)
(b)
(c)
(d)
(e)
the second is red given the first is red?
the second is a heart given the first is a club?
the second is the ace of clubs given the first is the ace of clubs?
the first card is red given the second card is red?
the second card is a heart given the first card is a heart if you
replace the first card after you draw it?
3. What is the probability that the sum of two dice is 9 given
that exactly one of the dice is a 4?
Cherveny
Oct 12
Math 1004: Probability
Practice Answers
1. Let E and F be events in sample space S. Suppose
P(E ) = 1/4, P(F ) = 1/2, and P(E ∪ F ) = 7/12. Calculate
(a) P(E ∩ F ) = 1/6
(b) P(E |F ) = 2/3
(c) P(F |E ) = 1/3
2. Two cards are drawn one after another from a standard 52
card deck without replacement. What is the probability..
(a)
(b)
(c)
(d)
(e)
25/51
13/51
0
25/51
1/4
3. P(sum 9|exactly one 4) = 1/5
Cherveny
Oct 12
Math 1004: Probability