So far they WON all matches,So will they LOSE the next ? ( The Gambler's Fallacy)

Well they won all the test matches against India. And so far all the matches of the CB series. They are not a very strong team as the previous Australian teams who controlled the world cricket arena for more than dozen years. But recently they have managed to win a fair amount of matches continuously.  Although Australia have the home advantage I think all the three teams have a somewhat fair chance of winning a match. The probability of an Australian win may be high but the probability of two consecutive wins will be low. And of course the probability of three continuous wins will be much lower. So Does this mean that Australia have a very high chance of loosing the next game ? or India have a high chance of winning the match on 12th ? or more importantly if Australia win on 12th Sri Lanka have very very high chance of winning the next match against Australia ?

This leads us to a very interesting fallacy in probability named as the Gambler's fallacy or Monte Carlo fallacy. So its time to think about this fallacy leaving cricket for a moment.

The Gambler's Fallacy

                 The basic idea of the gambler's fallacy is based on the perception we have that a result which we have obtained for a lengthy sequence will change in the next event. If we got 5 or more heads in a row while tossing an unbiased coin most of the time we will expect a tail in the next event. Not only we think but also we may incorrectly derive that the probability of a tail is much higher than 0.5 and reaching to 1. Lets check this using an example. Lets assume we have a fair coin so that the probability of a head or a tail equals 0.5. Lets consider the case where we get 5 consecutive heads. Then the probability will be 0.5^5=0.03125 which is a very rare event. So now consider a situation where we have already got 4 consecutive heads. So the big question is what is the probability of getting another head in the next tossing of the coin ? One may solve the problem in the following way.
Since the probability of getting 5 consecutive heads is 0.03125 the probability of not getting a head is 1-0.03125=0.96875. So the probabilities of getting a head and getting a tail is respectively 0.03125 and 0.96875. This reasoning highly favors a tail in the next tossing of the coin and this leads to the famous Gambler's Fallacy.
                            The problem of the above reasoning is that it does not take the independence of each tossing. It may be true that we had 4 consecutive heads so far. But it does not alter the probability of the next event. What has happened so far is simply the past and unlike in some real life scenarios they don't affect the future. There will be no demons or angels from the past events affecting the future when the events are simply independent.
                            So the probability of getting a head in the next tossing is still 0.5.  This confusion is the basic root of the gambler's fallacy. When a same result is observed for a large number of times people tend to think that the opposite will occur in the future. Based on that assumption they will bet on those results. This simple fallacy will lead to many interesting explanations of some events. Also there may be some instances where the gambler's fallacy may not be applied. As an example assumption of independence of events may not be totally correct. 
                             In the scenario at the beginning the events are not independent. There is a very slight dependence. Due to the consecutive wins they  may decide to give an opportunity to a new player or to rest some senior players. Then the Gambler's fallacy will not become a fallacy any more and the other two teams may increase their chance of winning. 

Merry Christmas and a Happy New Year !

It's Christmas.Though now it has become a commercial event, there is something special in every Christmas for me. In the early years of my life it was about the Santa and the presents I received. But the time has passed and now entering the period of my life where I have to give presents to others rather than expecting .Now its the time you see scary Santas in shops trying to sell various items.It's the time where you cant travel in a road even in the off peak hours.It's the time you see discounts everywhere.Anyway beside all the changes still its a time to rejoice.It's a time to enjoy with the family and friends.

This carol "Mistletoe and Wine" is a popular single by Cliff Richard.Written by Jeremy Paul, Leslie Stewart and Keith Strachan, this was originally performed as part of the musical Scraps which was based on Hans Christian Andersen's fairytale at the Orange Tree Theatre, Richmond, London in 1976.

The child is a King, the Caroller sing,
The old is past, there's a new beginning.
Dreams of Santa, dreams of snow,
Fingers numb, faces aglow.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts on the tree
A time to rejoice in the good that we see

A time for living, a time for believing
A time for trusting, not deceiving,
Love and laughter and joy ever after
Ours for the taking, just follow the master.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts in the tree
A time to rejoice in the good that we see

Silent night, holy night

Its a time for giving, a time for getting,
A time for forgiving and for forgetting.
Christmas is love, Christmas is peace,
A time for hating and fighting to cease.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts on the tree
A time to rejoice in the good that we see.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts on the tree
A time to rejoice in the good that we see.

Merry Christmas  and a Happy New Year !

Duckworth-Lewis Method (part 3)

1. Duckworth-Lewis Method (part 1)
2. Duckworth-Lewis Method (part 2)

After giving an introduction of Duckworth-Lewis method and some examples of Duckworth-Lewis method in the previous posts this post focuses on the  Duckworth-Lewis method and some of the previously used methods in rain affected games.

Before the invention of this method there have been several other methods used at the international level to decide the result of a rain affected match.One of them was the run rate based system. The main disadvantage of this method was that the number of wickets fallen was not considered. Thus in this system if a team has scored 100 runs for the lost of 9 wickets the position of the team is considered higher than scoring 90 runs for the loss of 1 wicket.

Eng v SA World Cup 1992
Situation at end of match SA need 22 runs.www.patrickeagar.com

Also another method that was used is the Highest Scoring Overs method which compares the maximum runs scored by team1 in any set of overs equal to the number of completed overs received by team 2 against the team 2 in those completed overs. So if team 2 received 31.4 overs their score after 31 overs is compared to the highest scoring 31 overs of team 1's innings. So this method becomes very unfair as the maidens bowled by the second team is not considered. So eventually this method is more biased towards the first team. The most controversial usage of this method arises in the 1992 world cup semi final between South Africa and England. Play was halted when South Africa had scored 231/6 from 42.5 overs against England's 252/6 from 45 overs. When play was able to resume there was time for South Africa to receive only one more ball, 43 overs in total, so their target was revised to 252 by discarding the two maiden overs in the England innings, one of which yielded one extra. So the number of bowls available was reduced by 12 but the target remained unchanged. Due to the use of this method an apparently attainable target suddenly became an impossible one. 


There are many other methods used in a rain affected match. All of these methods either uses the run rate , highest scoring overs or pre-calculated curves (as used in Duckworth-Lewis method) with slight variations. Yet Duckworth-Lewis method is regarded the best method so far used. But a slightly different method named as VJD system introduced by V.Jayadevan (an engineer from Thrissur in Kerala) has challenged the Duckwoth-Lewis method recently. 

THEN THEY CAME FOR ME..........

First they came for the communists,

and I didn't speak out because I wasn't a communist.

Then they came for the trade unionists,

and I didn't speak out because I wasn't a trade unionist.

Then they came for the Jews,

and I didn't speak out because I wasn't a Jew.

Then they came for me

and there was no one left to speak out for me.

 A famous statement attributed to pastor Martin Niemöller (1892–1984) 

How Secured is Your Security Question ????

You may have seen different kinds of questions such as "What is your library card no ?" , "what is your mothers birth town ?", while registering for an online account.These questions are used in password recovery processes and additional sign in verifications. Normally we all care about our passwords and worry about the security of the password.

But do we really care about these security questions ? It seems most of us don't care much about these.Most of the time we give our real details as the answers to theses questions and simply forget about the security question.Even I was unaware of security questions selected for my email address until i decided to write this post.The most surprising thing is that I can't come up with the correct answer for those questions.I think this is common between most of us.We really DON'T CARE about those.

But in reality this is a great security risk.Even if you have a very good password that can not be guessed easily,you may be vulnerable to security threats.It is like locking the front door while leaving the backdoor open.The confidentiality of the answers for a security question will be far less than a password.If some one asks you a password you probably wont tell.But if some one asks you the birth town of your mother you may answer it.Also most of the details that are asked in security questions can be found publicly or can easily be guessed.That is what happened with the email account of Sarah Palin the vice president candidate during 2008 elections.

A good security question should have a answer which can be easily memorized, which is not publicly available and which does not change overtime. Further the possible number of answers for the question should be very large so that it can not be guessed. There are some websites which offers good security questions. But it is worth to note that the goodness of the question depends on the expected answer as well.

Pigeon-Hole Principle

It's amazing how a very simple and obvious thing in life become a mathematical principle and helps to prove many complex results. Pigeon-Hole principle is one such concept. At first it seems as a very obvious fact and one may wonder whether it deserves such a place in mathematics. But this principle becomes very useful in many occasions to prove that there exists some answer for a problem.

Duckworth-Lewis Method (part 2)

(For an introduction of Duckworth-Lewis method visit the part 1)

Below, are two examples given for the betterment in understanding the Duckworth-Lewis method used in different scenarios.A sample of reduced D/L table is given below.

	Wickets lost
Overs left	0	2	5	7	9
50	100.0	83.8	49.5	26.5	7.6
40	90.3	77.6	48.3	26.4	7.6
30	77.1	68.2	45.7	26.2	7.6
25	68.7	61.8	43.4	25.9	7.6
20	58.9	54.0	40.0	25.2	7.6
10	34.1	32.5	27.5	20.6	7.5
5	18.4	17.9	16.4	14.0	7.0

Ex. 01: Team A scored 263 runs within their allotted 50 overs and the match was interrupted when team B had scored 132 for the loss of 2 wickets in 30 overs.

Case1: If match is not resumed.

The percentage of resources lost by Team B = 54% (wickets = 2 and overs left = 20)

The percentage of resources available to Team B = 100-54 = 46%

The percentage of Team A’s resources = 100% (no interruptions occurred during 1st innings) Revised target for 30 overs = 263*46/100 = 120.98

Since Team B has scored 132, they will be declared as the winners.

Case 2: If match is reduced to a 40 overs.

The percentage of resources at the time of interruption for Team B = 54%

The percentage of resources when match resumed (10 overs left, 2 wickets down) = 32.5% Percentage loss of resources for Team B = 54-32.5 = 21.5%

The percentage of resources available to Team B=100-21.5=78.5%

Revised target in 40 overs = 263*78.5/100 = 206.45 = 207(to win)

Ex2: Team A scored 132/2 in 30 overs and the match was interrupted. The match resumed as a 30 over match.

The percentage of resources at the time of interruption for Team A = 54%

The percentage of resources used by Team A = 100-54 = 46%

The percentage of resources available for Team B = 77.1% (30 overs 10 wickets)

Since Team 2 has more resources, their “revised target” must be raised upwards. The G50 value comes into play at this moment. The additional number of runs that has to be added to the target is calculated as a percentage of G50 value with respect to the additional resource percentage available for Team B.

The additional resource percentage available for team B = 77.1-46 = 31.1%

Additional runs that should be added to the target = 225*31.1/100 = 69.975 = 70 (225 is the G50 value)
Therefore Team B should score 202 runs to win the game within 30 overs.

So this shows how simple D/L method is used during matches.The professional version of this D/L method is used in ODI's.