Download What is probability?

What is probability?
Horse Racing
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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
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But ….

Some events cannot be repeated

In general, how can we find a probability of
an event?
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Gambling

The origin of modern probability theory
 Odds against an event A =  (賠率)
  = (1-P(A))/P(A)
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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
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If A occurs

We will win $ in the long run for a fair
game------ A game that is acceptable to both
sides.
 Why?
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Fair Game
- (1 – P(A)) +  P(A) = 0
 Because  P(A) = 1 – P(A)
 That is the game is fair to both sides

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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
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The equivalence between P(A)
and 
 = (1 – P(A)) / P(A)
 Conversely, P(A) = 1 / (1 + )

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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)?
 Odds=4/16=1/4
 P(A)=1/(1+1/4)=4/5
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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)
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For a fair coin

A---the occurrence of a head in one tossing
 Now P(A) = 0.5
  = (1 – P(A)) / P(A) = 1
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P(A) = 
=(1-)/
 If  > .5,  < 1
 If  < .5,  > 1

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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 + 

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The pari-mutuel system

A race with N horses (5 < N < 12)
 The bet on the i_th horse is B(i)
 We concern about which horse will win獨
贏
 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

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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?
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Reality
Track’s take t (0.17 < t < 0.185)
 If horse I wins, the payoff per dollar bet on
horse I, M(I) = B(1-t)/B(I)

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What is  ?

 = B(1-t)/B(I) - 1
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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 ?
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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.

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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.
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Past data

In Australia and the USA, favorite (大熱) or
near-favorite are “underbet” while longshots
(泠馬) are “overbet”.
 But it is not so in Hong Kong.
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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
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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%.
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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
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Which of the following is
correct?
(1)
(2)
(3)
(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)
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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.
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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)
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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
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Monty Hall Problem

Want to change your choice to the other
unopened door.
 Should you change?
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Improve your assessment

Given the occurrence of B, what is your
updated assessment of P?
 Answer--Bayes Theorem
 P(A|B)=P(B|A)P(A) /
(P(B|A)P(A)+P(~B|A)P(A))
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