What is probability

1
What is probability?

• Horse Racing

2
Relative Frequency

• Probability is defined as relative frequency
• When tossing a coin, the probability of getting a
  head is given by m/n
  
• Where n number of tossings
• m number of heads in n tossings

3
But .

• Some events cannot be repeated
• In general, how can we find a probability of an
  event?

4
Gambling

• The origin of modern probability theory
• Odds against an event A ? (??)
• ? (1-P(A))/P(A)

5
If A Does Not Occur

• We bet 1 on the occurrence of the event A
• If A does not occur, we lose 1
• In the long run, we will lose (1 P(A))
• Notice that we just ignore N, the number of the
  repeated games

6
If A occurs

• We will win ? in the long run for a fair
  game------ A game that is acceptable to both
  sides.
  
• Why?

7
Fair Game

• - (1 P(A)) ? P(A) 0
• Because ? P(A) 1 P(A)
• That is the game is fair to both sides

8
Interpretation of ?

• The amount you will win when A occurs assuming
  you bet 1 on the occurrence of A
  
• Gambling--- if ? is found and acceptable for
  both sides

9
The equivalence between P(A) and ?

• ? (1 P(A)) / P(A)
• Conversely, P(A) 1 / (1 ?)

10
Example

• Bet 16 on event A provided if A occurs we are
  paid 4 dollars (and our 16 returned) and if A
  does not occur we lose the 16. What is P(A)?
  
• Odds4/161/4
• P(A)1/(11/4)4/5

11
Is it arbitrary ?

• The axioms of probability
• (1) P(A) ?0
• (2) P(S)1 for any certain event S
• (3) For mutually exclusive events A and B,
• P(A ? B)P(A) P(B)

12
For a fair coin

• A---the occurrence of a head in one tossing
• Now P(A) 0.5
• ? (1 P(A)) / P(A) 1

13
P(A) ?

• ? ( 1 - ? ) / ?
• If ? .5, ?
• If ? 1

14
A First Prize of Mark Six

• Match 6 numbers out of 48
• P(A) (6/48) (5/47) (4/46) (3/45) (2/44) (1/43)
  1 / 12,271,512 8.15 x 10-8) .000,000,082
  
• In the past, when we have only 47 numbers,
• P(A) (6/47) (5/46) (4/45) (3/44) (2/43) (1/42)
  1 / 10,737,573 .000,000,09

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What is ? ?

• ? 12,271,511
• That is, you should win 12,271,511 for every
  dollar you bet
  
• Payoff 1 (bet) 12,271,511 (gain)
• In general, Payoff par dollar 1 ?

16
The pari-mutuel system

• A race with N horses (5
• The bet on the i_th horse is B(i)
• We concern about which horse will win??
• The total win pool B B(1) B(N)
• If horse I wins, the payoff per dollar bet on
  horse I M(I) B / B(I)

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What is ? ?

• ? B / B(I) - 1
• Let P(I) denote the winning probability of the
  horse I
  
• P(I) 1 / (? 1) B(I) / B
• That is the proportion of the bet on the horse I
  is the winning probability of the horse i

18
Implication

• The probability of winning can be reflected by
  the number B(I)/B
  
• Usually, B(I)/B fluctuates especially near the
  start of the horse racing
  
• Does this probability reflect the reality?

19
Reality

• Tracks take t (0.17
• If horse I wins, the payoff per dollar bet on
  horse I, M(I) B(1-t)/B(I)
  

20
What is ? ?

• ? B(1-t)/B(I) - 1

21
If I bet on the horse i

• Let p(I) denote the probability of the winning of
  the I_th horse
  
• If I lose, I will lose (1-p(I)) in the long
  run
  
• If I win, I will win p(I) (M(I) 1) in the
  long run
  
• What will happen if p(I) B(I) / B ?

22
If P(I) B(I) / B

• In the long run, I will gain p(I) (M(I) 1) 1
  t B(I) / B
  
• In the long run, I will lose (1 p(I)).
• So, altogether, I will lose t.

23
Objective probability

• From the record, we can group the horses with
  similar odds into one group and compute the
  relative frequency of the winners of each group
  
• We find the above objective probability is very
  close to the subjective probability B(I) / B.

24
Past data

• In Australia and the USA, favorite (??) or
  near-favorite are underbet while longshots (??)
  are overbet.
  
• But it is not so in Hong Kong.

25
Difficulty in assessing probability

• Example
• (1) Your patient has a lump in her breast
• (2) 1 chance that it is malignant
• (3) mammogram result the lump is malignant
• (4) The mammograms are 80 accurate for detecting
  true malignant lumps
  

26
Contd

• The mammogram is 90 accurate in telling a truly
  benign lumps
  
• Question 1 What is the chances that it is truly
  malignant?
  
• Ans. (1) less than .1 (2) less than 1 but
  larger than .1 (3) larger than 1 but less than
  50 (4) larger than 50 but less than 80 (5)
  larger than 80.

27
Accidence

• There were 76,577,000 flight departures in HK in
  the last two years (hypothetical)
  
• There were 39 fatal airline accidents (again,
  hypothetical)
  
• The ratio 39/76,577,000 gives around one accident
  per 2 million departures

28
Which of the following is correct?

• (1) The chance that you will be in a fatal plane
  crash is 1 in 2 million.
  
• (2) In the long run, about 1 out of every 2
  million flight departures end in a fatal crash
  
• (3) The probability that a randomly selected
  flight departure ends in a fatal crash is about
  1/(2,000,000)
  

29
Birthday

• How many people would need to be gathered
  together to be at least 50 sure that two of them
  share the same birthday?
  
• (1) 20 (2) 23 (3) 28 (4) 50 (5) 100.

30
Unusual hands in card games

• (1) 4 Aces, 4 Kings, 4 Queens and one spade 2.
• (2) Spade (A, K, 3) Heart (3, 4, 5) Diamond (A,
  2,4) Club (7, 8, 9, 10).
  
• Which has a higher probability
• Answer (1) (2)

31
Monty Hall Problem

• Three doors with one car behind one of the doors
• There are two goats behind the other two doors
• You choose one door
• Instead of opening the selected door, the host
  would open one of the other door with a goat
  behind it. Then he would ask if you

32
Monty Hall Problem

• Want to change your choice to the other unopened
  door.
  
• Should you change?

33
Improve your assessment

• Given the occurrence of B, what is your updated
  assessment of P?
  
• Answer--Bayes Theorem
• P(AB)P(BA)P(A) / (P(BA)P(A)P(BA)P(A))