1. Pitfalls of Probability
Great Theoretical Ideas In Computer Science
Pitfalls of Probability
Lecture 18 CS

2. You have 3 dice: Higher Number Wins6 2 7 A
2 Players each rolls a die. The player with the higher number wins 5 1 9 B
Which die is best to have – A, B, or C ? 4 3 8 C

3. A is better than B 6 2 7 5 1 9
When they are rolled, there are 9 equally likes outcomes
2 1
2 5
2 9
6 1
6 5
6 9
7 1
7 5
7 9
A beats B 5/9 of the time

4. B is better than C 5 1 9 4 3 8
When they are rolled, there are 9 equally likes outcomes
1 3
1 4
1 8
5 3
5 4
5 8
9 3
9 4
9 8
B beats C 5/9 of the time

5. A beats B with Prob. 5/9 B beats C with Prob. 5/9
QUESTION: Without explicitly doing the calculation, can you see the probability that A will beat C ?
QUESTION: If you chose first, which die would you take?
QUESTION: If you chose the second, which die would you take?

6. C is better than A! 6 2 7 4 3 8
When they are rolled, there are 9 equally likes outcomes
3 2
3 6
3 7
4 2
4 6
4 7
8 2
8 6
8 7
C beats A 5/9 of the time!

7. 6 2 7 4 3 8 5 1 9

8. First Moral
“Obvious” properties, such as transitivity, associativity, commutativity, etc… need to be rigorously argued because sometimes they are FALSE.

9. Second Moral
When reasoning about probabilities….
Stay on your toes!

10. Third Moral
To make money from a sucker in a bar, offer him the first choice of die. (allow him to change to your “lucky” die any time he wants)

11. Coming up next…
More of the pitfalls of probability…

12. A single summary statistic (such as an average) may have several different possible explanations…

13. Thrust of article: Market forces are at work
US News & World Report 1983
# of doctors
Average salary (1982 $)
1970
334,000
$103,900
1982
480,000
99,950
“Physicians are growing in number, but not in pay”
Thrust of article: Market forces are at work
Possibility:
Doctors earn more than ever, but many old doctors have retired and been replaced with younger ones.

14. Name a body part that almost everyone on earth had an above average number of
Fingers
Almost everyone has 10
More people are missing some than have extras (total fingers missing > # of extras)
Average: 9.99 …

15. Almost everyone can be above average!

16. Rare Disease
A person is selected at random and given a test for the rare disease “painanosufulitis”
Only 1/10,000 people have it
The test is 99% accurate: it gives the wrong answer (positive/negative) only 1% of the time
The Person tests POSITIVE
Does he have the disease?
What is the probability that he has the disease?

17. At most k/10,000 have the disease
Suppose there are k people in the population
At most k/10,000 have the disease
But k/100 have false test results
So  k/100 – k/10,000 have false test results but have no disease!
That’s about 100 times more likely he got a false positive.

18. Several years ago Berkeley faced a law suit …
% of make applicants admitted to graduate school was 10%
% of female applicants admitted to graduate school was 5%
Grounds for discrimination?
SUIT

19. Berkeley did a survey of its departments to find out which ones were at fault
The result was
SHOCKING…

20. Every Department is more likely to admit a female than a male
#of males accepted to department X
#of females accepted to department X
>
#of female applicants to department X
#of male applicants to department X

21. How can this be ?

22. Answer
Women tend to apply to departmebts that admit a smaller percentage of their applicants
Women
Men
Department
Applied
Accepted A 99 4 1 B 10
TOTAL
100 5

23. Let’s assume
3) Males and females have the same distribution of academic success

24. 1987 Department of Transportation requires that each month all airlines report their “on-time record”
# of on-time flights landing at nation’s 30 busiest airports
# of total flights into those airports

25. Newpapers would publish these data…
Meaningless junk!

26. Different airlines serve different airports with different frequency
An airline sending most of its planes into fair weather airports will crush an airline fling mostly into foggy airports
It can even happen that an airline has a better record at each airport, but gets a worse overall rating by this method.

27. American West B) Alaska Airlines
% on time
# flights LA
88.9
559
85.6
811
Phoenix
94.8
233
92.1
5255
San Diego
91.7
232
85.5
448 SF
83.1
605
71.3
449
Seattle
85.8
2146
76.7
262
OEVRALL
86.7
3775
89.1
7225
A beats B in each airport
B has a better overall rating!

28. Computer Performance Analysis
SYSTEM
Benchmark
6502
8080
Block
41.16
51.80
Sieve
63.17
48.08
SUM
104.33
99.58
Average
52.17
49.79
6502
as base
8080
Benchmark
Block
1.00
1.25
0.80
Sieve
0.76
1.31
SUM
2.00
2.01
2.11
Average
1.01
1.06
Ratio of total: 6502 takes 4.7% more than 8080
6502 as base: 6502 takes 1% less than 8080
8080 as base: 6502 takes 6% more than 8080

29. Patterson and Sequin (1982)
Code Size on RISC-I vs. Z8002
RISC AS
BASE
Benchmark
RISC
Z8002
E-String Search
140
126
1.0
0.9
F-Bit Test
120
180
1.5
H-Linked List
176
141
0.8
K-Bit Matrix
288
374
1.3
I-Quick Sort
992
1091
1.1
Ackerman(3,6)
144
302
2.1
Recursive 2Sort
2736
1368
0.5
Puzzle-S
2796
1398
0.51
Puzzle-P
752
602
SED
17720
Hanoi Tower 96
240
2.5
SUM
25960
23542
11.0
13.0
Average
2360
2140
1.18

30. A Bridge Hand has 13 cards. What distribution of the 4 suits is most likely?
Examples: ? ? ? ?

32. You walk into a pet shop. There are two parrots in a cage.
Shop owner says “At least one of the two is male”
What is the chance they are both male? FF
FM /3 chance they are both
MF male MM
Shop owner says “The dark one is male”
FM 1/2 chance they are both
MF male MM

33. Oct 93. Press Release Crime Crisis in Florida
9 foreign tourists killed that year in the state of Florida
Result:
German’s Advise their citizens not to visit Florida
All Indicative markings are removed from US. Rental Cars

34. Probability of  9 means in a given year
Suppose that each day there is a low probability p=1/365 of some foreign tourist dying somwhere in Florida (independent) (Thus, we get that the expected # of fatalities is 1 per year --- fairly low)
Probability of  9 means in a given year
Fairly low

35. Probability that in a given decade Florida will have a bloody year for foreign tourists:
1 - ( ) 10 = = 21%
The killing could easily be random fluctuations, not sufficient evidence to damage their economy and scare people

36. A Voting Puzzle
N (odd) people, each of whom has a random bit (50/50) on his/her forhead. No communication allowed. Each person goes to a private voting booth and casts a vote for either 1 or 0. If the outcome of the election coincided with the parity of the N bits, we say the voters “win” the election
How do voters maximize the probability of winning?

37. Notice: Each individual has no information about the parity
Beware of the Fallacy!
Since each individual is wrong half the time, the outcome of the election is wrong half the time
Cf. [BG] Random Oracles
PARITY – PR  PPR

38. Solution: NOTE – To know parity is equivalent to knowing the bit on your forehead
STRATEGY: Each person assumes the bit on his/her head is the same as the majority of bits sh/she sees. Vote accordingly (in the case of even split, vote 0).
ANALYSIS: The strategy works so long as the number of 1’s is not exactly one more than the number of 0’s.
Probability
Of winning

39. Optimal Strategy IF N = 1 mod 4 Vote your Bit
N = 3 mod 4 Vote opposite Bit
50% + 1/O(N) chance of winning
EX: N = 3
VOTE  1
Situations
All 0’s 3
Single 1
Two 1’s
Three 1’s
Correct 6/8 = 75% of the time

40. 0 | 0 1 | |
The guy in the first packet assumes he has a zero on his forehead. He votes accordingly.
If the other guys see that he indeed have a zero in his head, they assure his vote carries the election by splitting the vote in each packet.

41. 0 | 0 1 | |
The guys in the second packet assume the  of their two bits is zero.
If they are correct, the guys in the later packets slit their votes..

42. In general, if the people in a packet see that some lower packet has a bit-wise  of 0, they slit their vote. Otherwise they assume their packet ’s to 0 and vote accordingly.

43. New Chicago Puzzle
Each person has a private random bit
Now, what is their chance of winning the election?

44. Voting in Chicago
In Chicago people can vote as many times as they please
Strategy: each person behaves as a packet
Number the people from 1 to N. Person I votes 0 times if any lower number person has a 0 on his head. Otherwise person I assumes he has a 0 on his head and votes 2 times.
Correct with probability 1/2N
This is clearly optimal