Sandbox: Difference between revisions

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Mike Pearson provided a number of animations for these two articles. For the lottery example you can watch the numbers as they are chosen and produces a histogram for the number of times each number was chosen.   When it finishes it looks like the number 38 was chosen significantly more times than any other number. A second animations allows you to see gaps between each time a number is chosen.  You will see that the longest gap observed is 72 for the number 17 which seems surprising. The probability that a particular number would occur 72 times in a row is estimated to be about 1 in 12,5000.
dealt with what the British call League Tables but we would call rankings. David's [http://plus.maths.org/issue45/risk/ first column] appeared in Issue 44. Here David provides an example of a ranking where the outcomes should be pure chance but often do not look like pure chance.  In his [http://plus.maths.org/issue46/risk/index.html second column] which appeared in Issue 45, David provides an example where the outcomes should be  predoinantly skill also luck plays a role and he shows how to estimate how much of a role luck plays.  In this  the discussion is similar to our discussion of "Is poker a predominatly a game of skill or luck that we discussed in [http://chance.dartmouth.edu/chancewiki/index.php/Chance_News_29#Is_Poker_predominantly_skill_or_luck.3F Chance News 29]


The article then goes on to discuss how finding these apparent surprises does not necessarily indicate that the numbers were non randomly chosen. For the number of times a number occurs the authors show that the distribution for the number of times a number should occur in n draws has a binomial distribution with mean
In the UK National Lottery 6 numbers are randomly chosen from the numbers from 1 to 49. If you correctly predict the six numbers chosen you get a share of the jackpot.
 
<center> http://www.dartmouth.edu/~chance/forwiki/clt.jpg</center>
 
They use a similar analysis to show that a gap of 72 is not too surprising. However, here they change the question.They calculate that in 1240 draws, each having 6 number, there are 7440 gaps that aspear. Then they ask:"What is the chance that the longest of these 7440 gaps is at lest 72". Using this they find there is about a 50% chance that a gap of at least 72 would appear. 
 
Finally they discuss the use of a Chi Squared statistic to see if the approximate distributions from the data are consistent with the theoretical distributions used.

Revision as of 20:20, 13 March 2008

dealt with what the British call League Tables but we would call rankings. David's first column appeared in Issue 44. Here David provides an example of a ranking where the outcomes should be pure chance but often do not look like pure chance. In his second column which appeared in Issue 45, David provides an example where the outcomes should be predoinantly skill also luck plays a role and he shows how to estimate how much of a role luck plays. In this the discussion is similar to our discussion of "Is poker a predominatly a game of skill or luck that we discussed in Chance News 29

In the UK National Lottery 6 numbers are randomly chosen from the numbers from 1 to 49. If you correctly predict the six numbers chosen you get a share of the jackpot.

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