Read Online 100 Essential Things You Didn't Know You Didn't Know by John D. Barrow - Free Book Online
Authors: John D. Barrow
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don’t think that random sequences can have all those long runs of heads or tails, but their presence is one of the acid tests for the genuineness of a random sequence of heads and tails. The coin tossing process has no memory. The chance of a head or a tail from a fair toss is 1/2 each time, regardless of the outcome of the last toss. They are all independent events. Therefore, the chance of a run of r heads or r tails coming up in sequence is just given by the multiplication ½ × ½ × ½ × ½ × . . . × ½, r times. This is ½ r . But if we toss our coin N times so that there are N different possible starting points for a run of heads or tails, our chance of a run of length r is increased to N × ½ r . A run of length r is going to become likely when N × ½ r is roughly equal to 1 – that is, when N = 2 r . This has a very simple meaning. If you look at a list of about N random coin tosses, then you expect to find runs of length r where N = 2 r .
All our six sequences were of length N = 32 = 2 5 so if they are randomly generated we expect there is a good chance that they will contain a run of 5 heads or tails and they will almost surely contain runs of length 4. For instance, with 32 tosses there are 28 starting points that allow for a run of 5 heads or tails, and on average two runs of each is quite likely. When the number of tosses gets large, we can forget about the difference between the number of tosses and the number of starting points and use N = 2 r as the handy rule of thumb. fn1 The absence of these runs of heads or tails is what should make you suspicious about the first three sequences and happy about the likely randomness of the second three. The lesson we learn is that our intuitions about randomness are biased towards thinking it is a good deal more ordered than it really is. This bias is manifested by our expectation that extremes, like long runs of the same outcome, should not occur – that somehow those runs are orderly because they are creating the same outcome each time.
These results are also interesting to bear in mind when you look at long runs of sporting results between teams that regularly play one another, Army vs Navy, Oxford vs Cambridge, Arsenal vs Spurs, AC Milan vs Inter, Lancashire vs Yorkshire. There are often winning streaks where one team wins for many years in a row, although this is not usually a random effect: the same players form the core of the team for several years and then retire or leave and a new team is created.
fn1 The result is easily generalised to deal with random sequences where the equally likely outcomes are more than two (H and T here). For the throws of a fair die the probability of any single outcome is 1/6 and to get a run of the same outcome r times we would expect to have to throw it about 6 r times; even for small values of r, this is a very large number.
25
The Flaw of Averages
Statistics can be made to prove anything – even the truth.
Noël Moynihan
Averages are funny things. Ask the statistician who drowned in a lake of average depth 3 centimetres. Yet, they are so familiar and seemingly so straightforward that we trust them completely. But should we? Let’s imagine two cricketers. We’ll call them, purely hypothetically, Flintoff and Warne. They are playing in a crucial test match that will decide the outcome of the match series. The sponsors have put up big cash prizes for the best bowling and batting performances in the match. Flintoff and Warne don’t care about batting performances – except in the sense that they want to make sure there aren’t any good ones at all on the opposing side – and are going all out to win the big bowling prize.
In the first innings Flintoff gets some early wickets but is taken off after a long spell of very economical bowling and ends up with figures of 3 wickets for 17 runs, an average of 5.67. Flintoff’s side then has to bat, and Warne is on top form, taking a succession of wickets for final
All our six sequences were of length N = 32 = 2 5 so if they are randomly generated we expect there is a good chance that they will contain a run of 5 heads or tails and they will almost surely contain runs of length 4. For instance, with 32 tosses there are 28 starting points that allow for a run of 5 heads or tails, and on average two runs of each is quite likely. When the number of tosses gets large, we can forget about the difference between the number of tosses and the number of starting points and use N = 2 r as the handy rule of thumb. fn1 The absence of these runs of heads or tails is what should make you suspicious about the first three sequences and happy about the likely randomness of the second three. The lesson we learn is that our intuitions about randomness are biased towards thinking it is a good deal more ordered than it really is. This bias is manifested by our expectation that extremes, like long runs of the same outcome, should not occur – that somehow those runs are orderly because they are creating the same outcome each time.
These results are also interesting to bear in mind when you look at long runs of sporting results between teams that regularly play one another, Army vs Navy, Oxford vs Cambridge, Arsenal vs Spurs, AC Milan vs Inter, Lancashire vs Yorkshire. There are often winning streaks where one team wins for many years in a row, although this is not usually a random effect: the same players form the core of the team for several years and then retire or leave and a new team is created.
fn1 The result is easily generalised to deal with random sequences where the equally likely outcomes are more than two (H and T here). For the throws of a fair die the probability of any single outcome is 1/6 and to get a run of the same outcome r times we would expect to have to throw it about 6 r times; even for small values of r, this is a very large number.
25
The Flaw of Averages
Statistics can be made to prove anything – even the truth.
Noël Moynihan
Averages are funny things. Ask the statistician who drowned in a lake of average depth 3 centimetres. Yet, they are so familiar and seemingly so straightforward that we trust them completely. But should we? Let’s imagine two cricketers. We’ll call them, purely hypothetically, Flintoff and Warne. They are playing in a crucial test match that will decide the outcome of the match series. The sponsors have put up big cash prizes for the best bowling and batting performances in the match. Flintoff and Warne don’t care about batting performances – except in the sense that they want to make sure there aren’t any good ones at all on the opposing side – and are going all out to win the big bowling prize.
In the first innings Flintoff gets some early wickets but is taken off after a long spell of very economical bowling and ends up with figures of 3 wickets for 17 runs, an average of 5.67. Flintoff’s side then has to bat, and Warne is on top form, taking a succession of wickets for final
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