Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of...
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![Page 1: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/1.jpg)
Conditional Probability
![Page 2: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/2.jpg)
Conditional Probability A newspaper editor has 120 letters
from irate readers about the firing of a high school basketball coach.
The letters are divided among parents and students, in support of or against the coach
They have space to print only one of these letters.
![Page 3: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/3.jpg)
Conditional Probability The break down of the letters:
What are the chances that a student letter supporting the coach will be chosen?
Written by students
Written by parents
Total
Support coach 16 44 60
Against coach 8 52 60
Total 24 96 120
![Page 4: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/4.jpg)
Conditional Probability
Let’s look at a Venn Diagram: Let C: event the letter is from a student Let T: event the letter favors the coach
C T
120
16
120
8
120
44
120
52 CTCP
CCTP
TCP
CCTP
![Page 5: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/5.jpg)
Conditional Probability From the Venn Diagram:
Slim chance a student letter supporting the coach will be printed:
Could be unfair: student letters support the coach by a ratio of 2 : 1
This fact is evident since
133.0120
16TCP
3
2
20.0
133.0
CP
CTP
![Page 6: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/6.jpg)
Conditional Probability
What does tell us?
Given the letter came from a student, the chance it supports the coach is two-thirds
In other words: 20% of the letters came from students. Of those, two-thirds were in favor of the coach
3
2
20.0
133.0
CP
CTP
![Page 7: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/7.jpg)
Conditional Probability Notice previous Venn Diagram probabilities
were all relative to sample space:
For example:
looks at probability a letter supports a teacher based on a reduced sample space, student letters only
CPCTP
120
16CTP
3
2
24
16
12024
12016
CP
CTP
![Page 8: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/8.jpg)
Conditional Probability What does this mean?
Knowing some info beforehand can change a probability
Ex: Probability of rolling a 12 with 2 dice is 1/36, but if you know the first die is a 4, the probability is 0. If the first die is a 6, the probability is 1/6
Determining a probability after some information is known is called conditional probability
![Page 9: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/9.jpg)
Conditional Probability
Notation means the probability of E
happening given that F has already occurred
Definition
FEP |
0 where, |
FPFP
FEPFEP
This is a conditional probability
![Page 10: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/10.jpg)
Conditional Probability
The formula implies:
|FP
FEPFEP
FEPFPFEP |
EFPEPFEP |
Notice the reversal of the events E and F
Note: EFPFEP || Very Important!These are two
different things. They aren’t always equal.
![Page 11: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/11.jpg)
Conditional Probability
Ex: Suppose 22% of Math 115A students plan to major in accounting (A) and 67% on Math 115A students are male (M). The probability of being a male or an accounting major in Math 115A is 75%. Find and .
MAP | AMP |
![Page 12: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/12.jpg)
Conditional Probability
Sol:
First find
MPMAP
MAP
|
MAP
14.0
75.067.022.0
MAPMPAPMAP
![Page 13: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/13.jpg)
Conditional Probability
Sol:
2090.067.0
14.0
|
MP
MAPMAP
![Page 14: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/14.jpg)
Conditional Probability
Sol:
6364.022.0
14.0
|
AP
AMPAMP
![Page 15: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/15.jpg)
Conditional Probability Sometimes one event has no effect
on another Example: flipping a coin twice
Such events are called independent events
Definition: Two events E and F are independent if or EPFEP | FPEFP |
![Page 16: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/16.jpg)
Conditional Probability
Implications:
FPEPFEP
EPFP
FEP
EPFEP
|
So, two events E and F are independent if this is true.
![Page 17: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/17.jpg)
Conditional Probability
The property of independence can be extended to more than two events:
assuming that are all independent.
nn EPEPEPEEEP 2121
nEEE ,,, 21
![Page 18: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/18.jpg)
Conditional Probabilities
INDEPENDENT EVENTS AND MUTUALLY EXCLUSIVE EVENTS ARE NOT THE SAME
Mutually exclusive:
Independence:
0FEP
FPEPFEP
EPFEP
|
![Page 19: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/19.jpg)
Conditional Probability
Ex: Suppose we roll toss a fair coin 4 times. Let A be the event that the first toss is heads and let B be the event that there are exactly three heads. Are events A and B independent?
TTTTTTTHTTHTTTHH
THTTTHTHTHHTTHHH
HTTTHTTHHTHTHTHH
HHTTHHTHHHHTHHHH
S
,,,
,,,,
,,,,
,,,,
![Page 20: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/20.jpg)
Conditional Probability
Soln:For A and B to be independent,
and
Different, sodependent
TTTTTTTHTTHTTTHH
THTTTHTHTHHTTHHH
HTTTHTTHHTHTHTHH
HHTTHHTHHHHTHHHH
S
,,,
,,,,
,,,,
,,,,
21
168 AP 4
1164 BP
1875.0163 BAP
BPAPBAP
125.081
41
21 BPAP
![Page 21: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/21.jpg)
Conditional Probability
Ex: Suppose you apply to two graduate schools: University of Arizona and Stanford University. Let A be the event that you are accepted at Arizona and S be the event of being accepted at Stanford. If and , and your acceptance at the schools is independent, find the probability of being accepted at either school.
7.0AP 2.0SP
![Page 22: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/22.jpg)
Conditional Probability
Soln: Find .
Since A and S are independent,
SAP
SAPSPAPSAP
14.0
2.07.0
SPAPSAP
![Page 23: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/23.jpg)
Conditional Probability
Soln:
There is a 76% chance of being accepted by a graduate school.
76.0
14.02.07.0
SAPSPAPSAP
![Page 24: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/24.jpg)
Conditional Probability
Independence holds for complements as well.
Ex: Using previous example, find the probability of being accepted by Arizona and not by Stanford.
![Page 25: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/25.jpg)
Conditional Probability
Soln: Find .
56.0
8.07.0
2.017.0
CC SPAPSAP
CSAP
![Page 26: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/26.jpg)
Conditional Probability
Ex: Using previous example, find the probability of being accepted by exactly one school.
Sol: Find probability of Arizona and not Stanford or Stanford and not Arizona.
CC ASSAP
![Page 27: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/27.jpg)
Conditional Probability
Sol: (continued)Since Arizona and Stanford are mutually exclusive (you can’t attend both universities)
(using independence)
CCCC ASPSAPASSAP
CC APSPSPAP
![Page 28: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/28.jpg)
Conditional Probability
Soln: (continued)
62.0
06.056.0
3.02.08.07.0
CC
CCCC
APSPSPAP
ASPSAPASSAP
![Page 29: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/29.jpg)
Conditional Probability
Independence holds across conditional probabilities as well.
If E, F, and G are three events with E and F independent, then
GFPGEPGFEP |||
![Page 30: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/30.jpg)
Conditional Probability
Focus on the Project: Recall: and
However, this is for a general borrower
Want to find probability of success for our borrower
464.0SP 536.0FP
![Page 31: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/31.jpg)
Conditional Probability
Focus on the Project: Start by finding and
We can find expected value of a loan work out for a borrower with 7 years of experience.
YSP | YFP |
![Page 32: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/32.jpg)
Conditional Probability Focus on the Project:
To find we use the info from the DCOUNT function
This can be approximated by counting the number of successful 7 year records divided by total number of 7 year records
YSP |
YPYSP
YSP
|
![Page 33: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/33.jpg)
Conditional Probability
Focus on the Project: Technically, we have the following:
So,
BR
BRBRBRBR YP
YSPYSPYSP
||
4393.0| 239105 YSP
Why “technically”? Because we’re assuming that the loan workouts BR bank made were made for similar types of borrowers for the other three. So we’re extrapolating a probability from one bank and using it for all the banks.
![Page 34: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/34.jpg)
Conditional Probability
Focus on the Project: Similarly,
This can be approximated by counting the number of failed 7 year records divided by total number of 7 year records
YPYFP
YFP
|
![Page 35: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/35.jpg)
Conditional Probability
Focus on the Project: Technically, we have the following:
So,
BR
BRBRBRBR YP
YFPYFPYFP
||
5607.0| 239134 YFP
![Page 36: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/36.jpg)
Conditional Probability
Focus on the Project: Let be the variable giving the value of a loan work out for a borrower with 7 years experience
Find
YZ
YZE
![Page 37: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/37.jpg)
Conditional Probability
Focus on the Project:
This indicates that looking at only the years of experience, we should foreclose (guaranteed $2.1 million)
000,897,1$
5607.0000,2504393.0000,000,4
Failure Prob. Failure Success Prob. Success
YZE
![Page 38: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/38.jpg)
Conditional Probability
Focus on the Project: Of course, we haven’t accounted for the other two factors (education and economy)
Using similar calculations, find the following:
CFPCSPTFPTSP | and,|,|,|
![Page 39: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/39.jpg)
Conditional Probability
Focus on the Project:
5581.0| 1154644 TFP
4419.0| 1154510 TSP
5217.0| 1547807 CSP
4783.0| 1547740 CFP
![Page 40: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/40.jpg)
Conditional Probability
Focus on the Project: Let represent value of a loan work out for a borrower with a Bachelor’s Degree
Let represent value of a loan work out for a borrower with a loan during a Normal economy
TZ
CZ
![Page 41: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/41.jpg)
Conditional Probability
Focus on the Project: Find and TZE CZE
000,907,1$
5581.0000,2504419.0000,000,4
Failure Prob. Failure Success Prob. Success
TZE
000,206,2$
4783.0000,2505217.0000,000,4
Failure Prob. Failure Success Prob. Success
CZE
![Page 42: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/42.jpg)
Conditional Probability
Focus on the Project: So, two of the three individual
expected values indicates a foreclosure:
000,897,1$YZE
000,206,2$CZE
000,907,1$TZE
![Page 43: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/43.jpg)
Conditional Probability Focus on the Project:
Can’t use these expected values for the final decision
None has all 3 characteristics combined:
for example has all education levels and all economic conditions included
YZE
![Page 44: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/44.jpg)
Conditional Probability Focus on the Project:
Now perform some calculations to be used later
We will use the given bank data:That is is reallyand so on…
SCPSTPSYP | and,|,|
SYP | BRBR SYP |
![Page 45: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/45.jpg)
Conditional Probability
Focus on the Project: We can find
since Y, T, and C are independent
Also
SCPSTPSYPSCTYP |||| SCTYP |
FCPFTPFYPFCTYP ||||
![Page 46: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/46.jpg)
Conditional Probability
Focus on the Project:
Similarly:
0714.01470
105
in number
and in number
||
BR
BRBR
BRBR
S
SY
SYPSYP
5823.01386
807| SCP
5301.0962
510| STP
![Page 47: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/47.jpg)
Conditional Probability
Focus on the Project:
0220.0
5823.05301.00714.0
||||
SCPSTPSYPSCTYP
![Page 48: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/48.jpg)
Conditional Probability
Focus on the Project:
0753.01779
134| FYP
5222.01417
740| FCP
5314.01212
644| FTP
![Page 49: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/49.jpg)
Conditional Probability
Focus on the Project:
0209.0
5222.05314.00753.0
||||
FCPFTPFYPFCTYP
![Page 50: Conditional Probability. A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach. The letters are.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d495503460f94a2596f/html5/thumbnails/50.jpg)
Conditional Probability
Focus on the Project:
Now that we have found and we will use these values to find and
SCTYP | FCTYP |
CTYSP | CTYFP |