Download + P(B)

Section 6.2: Probability Models
Ways to show a sample space of outcomes of multiple actions/tasks:
(example: flipping a coin and rolling a 6 sided die)
2) Multiplication (or
1) Tree diagram
Counting) Principle
If the 1st task can happen n
ways and the 2nd can
happen m ways, then are
n•m outcomes of the task
together.
Task #1 ↑
Task #2 ↑
So there 2 results on a coin
and 6 results for a die, so
there are 6 • 2 = 12 possible
outcomes together .
3) Listing systematically (organized list)
(for the example of flipping a coin and rolling a die)
List all the possible outcome that result in heads followed by all
the outcomes that result in tails
OR
all the outcomes that have a roll of 1, followed by 2, and so on.
Statistics (specifically probability) is not all about
flipping coins and rolling dice, but the outcomes
can simulate many real world random
phenomenon.
If selecting objects from a collection of distinct choices (ex:
drawing playing cards), there are two important situations that
must be considered:
Selecting WITH replacement:
Selecting WITHOUT replacement:
Choose, record, put back
Choose, record, keep
How many ways can a 3 digit numbers be made?
Using the counting principle:
Using the counting principle:
10 10 10 = 1000 ways
____•____•____
1st 2nd 3rd
digit digit digit
10 9
8 = 720 ways
____•____•____
1st 2nd 3rd
digit digit digit
A permutation of n objects using only r of those objects in
Pr.
each arrangement is written as n
SOME of the objects.
This is if you only use
In a race with eight horses, how many ways can 1st, 2nd, and
3rd place be awarded?
8P 3
= 336
This can also be done using the counting principle…
8* 7
1st
2nd
*
6 = 336
3rd
A combination is an arrangement where the order does not
matter (ABC is the same as BCA). A combination of n object
taken r at a time is symbolized as nCr.
Example: A state’s department of transportation plans to
develop a new section of interstate highway and receives 16
bids for the project. The state plans to hire 4 of the bidding
companies. How many different combinations of the 4
companies can be selected?
16C4 = 1820
Rules of Probability
1) 0 ≤ P(E) ≤ 1
2) P(S) = 1 {S represents the entire sample space of
outcomes}
3) If events A & B are disjoint (cannot BOTH happen in
one trial) P(A or B) = P(A) + P(B)
4) P(not E) = 1 – P(E)
5) If events A & B are independent, P(A and B) = P(A) •
Set
P(B)Notation for Probability:
A  B = A or B
A  B = A and B
Ec =
not E
When events are not disjoint (can happen simultaneously),
the probability of their union (A or B) is calculated as follows:
P(A or B) = P(A) + P(B) – P(A and B)
or
P(A U B) = P(A) + P(B) – P(A ∩ B)
Example:
55% of adults drink coffee
25% of adults drink tea
45% of adults drink cola
15% drink both coffee and tea
5% drink all three
25% drink coffee and cola
5% drink only tea
What percent drink cola and tea?
What percent drink NONE of these?
T
Cf
20
10
5
20 5 5
15
Cl
20