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Customer Review

36 of 45 people found the following review helpful
3.0 out of 5 stars Enjoyable book - but watch out for the error in Chapter 1., 5 July 2012
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This review is from: Paradox: The Nine Greatest Enigmas in Physics (Kindle Edition)
I found this book both interesting and enjoyable. The discussion on the implications of Einstein's theory was well presented. However I have to warn you about an error in Chapter 1 of the book which I will highlight briefly below.

To explain how knowledge affects probabilities the author gives the following example. Suppose you call a pet shop and tell them that you want to buy two male kittens. The owner tells you that he has just taken delivery of two kittens, a black and a tabby. If he tells you no more, then the chance that they are both males is 1 in 4 (assuming 50% of all kittens are male). If the owner tells you he has looked at the tabby and it's a male, then the odds of two males becomes 50%. All fine so far, but here comes the mistake. According to the author, if the pet shop owner tells you he has looked at one of the kittens and found it to be a male, but does not tell which one, then the odds of two males is 1 in 3 or 33%.

In fact, as soon as you know that at least one of the two kittens is male, the odds of two males becomes 50%, whether or not you know which one the pet shop owner checked. I would have thought such an error would have been spotted before the book was published and it did make me wonder whether I could rely on the more complicated explanations in the rest of the book. I also emailed the Professor pointing out what I believe is a mistake in the book, but so far, he has not replied.

Had it not been for the mistake I would have scored this book more highly. It is well written and I found it enjoyable and interesting.

Update - Since writing the orginal review I have now had email correspondence with Pof. Al-Khalili. He has agreed that the example in chapter 1 is incorrect and he intends to get in touch with his publishers to correct future editions of the book.
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Showing 1-10 of 33 posts in this discussion
Initial post: 17 Jul 2012 11:37:41 BDT
Last edited by the author on 17 Jul 2012 11:38:42 BDT
Book Boy says:
Just a comment on the alleged error in Chapter 1 -

If no information is given, then there are four possibilities - (male,male) (female,female) (black male, tabby female) & (black female, tabby male) and hence the chances of both being male is 1 in 4.

If it is stated that at least one of the kittens is male, then there are three possibilities - (male,male) (black male, tabby female) & (black female, tabby male) and hence the chances of both being male is 1 in 3.

If it is stated that say the black kitten is male, then there are two possibilities - (male,male) & (black male, tabby female) and hence the chances of both being male is 1 in 2.

I have to admit that I haven't read / bought the book yet - which was why I was reading the reviews - so I don't know how the author tried to explain the solution to the problem. Perhaps I should wait for further reviews before deciding to buy?

In reply to an earlier post on 18 Jul 2012 11:48:19 BDT
Alex B says:
The author explains his assertion that the chances of two males is 1 in 3 just as you said. i.e. there are 3 possibilities left and only 1 is male, male. In fact since you don't know which cat is male but you know 1 is male the possibility of black male, tabby female is the same as the possbility of tabby male, black female and is 25%.

If you like I could give a more detailed explanation of why this is the case. Overall though, I did enjoy the book.

In reply to an earlier post on 25 Jul 2012 12:37:44 BDT
Book Boy says:
I appreciate that the argument you made for the odds of getting 2 male kittens being 50% (even when you are only told that at least one of the kittens is male) is quite convincing - but I also find the other argument (which suggests that the odds are 33.33% for two male kittens) equally so.

You may be interested to read page 163 of "Taking Chances" by John Haigh, which discusses this paradox (admittedly with children rather than kittens).

It seems to me that whilst the various arguments seem convincing - they don't actually provide a rigorous proof - I believe this can only be achieved by running some sort of simulation process.

The simplest way of doing this would be to flip a coin where, say, heads would represent a female kitten and tails a male. One simulation run would consist of two coin flips, which could have the outcome of:
[A] two tails (i.e. male:male)
[B] heads/tails or tails/heads (i.e. female:male or male:female) or
[C] two heads (i.e. female:female).
In case [A], 1 would be added to the male:male total, in the case of [B], 1 would be added to the (female:male / male:female) total and in case [C] nothing would be added to either total.

The problem with this simple simulation is that it would be extremely tedious and time-consuming, to get sufficient results to arrive at a statistically valid result. Which is why I decided to write a short BASIC program to generate the data.

Four successive runs of the program with 1,000 iterations (equivalent to 1,000 pairs of coin flips) per run, generated the following results:-

Male:Male 249 274 280 227
Male:Female / Female:Male 520 498 475 509
Female:Female 231 228 245 264
% Male:Male 32.4 35.5 37.1 30.8

The average male:male percentage for the data from all four runs is 33.95% which suggests that the argument that proportion of male:male kittens is 1 in 3 rather than the alternative 1 in 2 (or 50%).

[If the random numbers generated by the program really are random - it would be expected that 25% of the sets of numbers generated in each run would be rejected as corresponding to the disallowed female:female case. From the above table it can be calculated that the total "female:female cases" (i.e. 968) represents 24.2% of the 4,000 iterations which is close to the 25% ideal].

In reply to an earlier post on 25 Jul 2012 14:00:16 BDT
Alex B says:
Thanks for your comment. I may have a look at that book you mentioned - "Taking Chances" - it sounds interesting. I had a very similar discussion with a friend of mine on this point and he also introduced the idea of looking at coin tosses, instead of cats. I eventually convinced him that my point of view was the correct one, although it did take quite a few emails before he was convinced.

As I see it the problem with the simulation is that it does not truly replicate the example outlined in the book. Let me try to explain why I think that. By starting with a random sample and discounting all the female/female or head/head occurrences you have altered the odds. You have started with a truly random sample in which two tails would be 25% of the total and then discarded 25% of the total sample and so the proportion of tail/tail occurrences to the remaining total is bound to be 1 in 3.

It seems to me the equivalent in the pet shop example of the that coin tossing simulation would be the following. You phone up the pet shop owner and he tells you that he can guarantee that the cats will not be two females. You ask him how he can guarantee that and he says that he will look at both cats and if they are both females, he will send them back and get another two cats. That is not the example in the book.

The example in the book is that the pet shop owner has looked at only one of the cats and found what sex it is. The question is then what is the chance that both cats have the same sex as the one he has already looked at. In the example in the book, the sex of the first cat looked at was male and the question was - Given that one of the cats is a male, what is the probability of two males? A completely equivalent example would be if he looked at the first cat and found it was female and you were then asked what is the probability that both cats would be female. As you know, I say in both cases it would be 50%.

Here is another example with coin tosses which may help to convince you.

A) Suppose I toss two identical coins and cover them so I can't see if they came up heads or tails. I then look at one of them and let's say it is a head. I tell you that I tossed two identical coins and looked at one and found it was a head. If I then ask you what is the probability that both coins would be heads, I'm fairly sure you would say 50% (as would I hope most people with any intelligence.)

B) I repeat the example but this time the two coins are not identical. Let's say one coin is a £1 coin and the other is a 10p coin. I then look at one of the coins and let's say this time it is a tail. I tell you that I have looked at one of the coins and found it is a tail and ask you what is the chance that both coins will be tails. Again I'm pretty sure you would say 50% .

C) I then tell you that the second time the coins were not identical although they are both true coins with a 50% chance of being a head or a tail. I ask you if that knowledge would change your estimate of both coins being tails. Hopefully you would say no, it is still 50%.

You see to accept the argument that Professor Al-Khalili has in his book Paradox, it seems to me you would have to believe the following with regard to the example above. - I tell you that I tossed two coins; that one coin was a £1 and the other a 10p and that I looked at one of the coins and told you what it was (Let's say that it is a head on this occasion.) I then pose the question - what is the chance that both coins will be heads?. According to Professor Al-Khalili it would be 50% if I tell you which coin I looked at and 33% if I don't. I cannot agree.

In reply to an earlier post on 28 Jul 2012 17:30:29 BDT
If you flick both coins and ask the question what is the possibility of both coins landing heads then the answer is 33% as there are only three possible combinations available to us Tail/Tail or Tail/Head or Head/Head.

If the prof tells you that one of the coins has landed on heads then the probability ratio for that coin collapses as the probability of it being heads is a 100% and the coin can be factored out of the equation, which leaves only the probability of the other coin to be ascertained which must be 50% as it outcome is totally chance. Telling you which coin he looked at only decides which coin is left for analysis and for either one the probability is 50%.

Saying that... there is a gameshow puzzle involving goats in a box which I dimly remember published in Penthouse in the 1970's which flummoxed some of the finest mathematicians in the USA so maybe I'm missing something...maybe it's a Schrodinger's catterism which is altering the outcome purely by observing the coin?

In reply to an earlier post on 29 Jul 2012 00:10:52 BDT
I. GRIFFITHS says:
Try looking at the ''Monty Hall Problem'' ?

this may help you

In reply to an earlier post on 29 Jul 2012 12:22:23 BDT
Alex B says:
Just to be clear - let's set out the two coin toss probabilities in full.

If you flick two coins and ask the probability of two heads, it is 25% (not 33%). All four possibilities are equally likely. ;-
tail/head (i.e. the 1st coin tossed is head and the 2nd a tail)
tail/tail,
head/tail, and
head/head

Suppose you are then told one of the coins is a head but you are not told which coin it is. Then the probability of two tails becomes zero, two heads becomes 50% and tail/head and head/tail are 25% for each.

Let's say you are then told the first coin was the one that was found to be a head. On obtaining this extra piece of information the probability of tail/head becomes zero, the probability of head/tail becomes 50% and head/head remains at 50%.

PS - The professor does not talk about coin tosses in the book - this has just emerged as another example from discussion of my review.

In reply to an earlier post on 29 Jul 2012 12:29:39 BDT
Alex B says:
I did look at the Monty Hall problem, which is correctly described. However the example with the kittens is different and is unfortunately incorrect in my opinion.

The author is correct in saying that there are still three possibilities if you are only told that one of the kittens is a boy. If you are not told which of the kittens is a boy then for you the probability of black boy/tabby female will be equal to the probability of black female/tabby boy and will be 25%., with the probability of two boys being 50%.

In reply to an earlier post on 29 Jul 2012 17:51:34 BDT
Last edited by the author on 29 Jul 2012 17:53:17 BDT
Book Boy says:
Thanks for your detailed reply to my earlier post - I have had another thought - which might just reconcile our apparently different interpretations of the kitten problem.

It seems to me that so far we have failed to take into account the fact that different observers can experience different odds whilst appearing to observe the same problem. As an example consider the three-door / Monty Hall problem -

Suppose a member of the audience arrives late - i.e. just in time to see that one of the doors has been opened but before the contestant has made their final choice. As we know, the contestant will double their chances of winning if they change their original choice, (from 1 in 3 to 2 in 3) but if the late arrival was allowed to choose instead, their chances of getting the valuable prize would only be 1 in 2. This is because they would have a 50% chance of choosing the door that has a 1 in 3 chance of concealing the prize and a 50% chance of selecting the door that has a 2 in 3 chance of concealing the prize - 50% of (1/3 + 2/3) equivalent to an overall 50% chance of winning. So although they appear to be looking at the same problem, their chances of succeeding are different, because one of them has some prior knowledge that is denied to the other.

Returning to the kittens - when the pet-shop owner checks one of the kittens and finds that it is a male, he can't help but know which one it is that he has checked and therefore his chance of guessing whether or not the second kitten is also a male has got to be 50%. However, if he merely informs the potential customer that one of the kittens is male, the customer has only a 33.33% chance of getting two male kittens if he decides to purchase them. This is because the customer is lacking some of the information known to the pet-shop owner.

Perhaps it boils down to a bit of Bayesian statistics?

In reply to an earlier post on 30 Jul 2012 00:13:57 BDT
Alex B says:
Thanks for your reply. I agree with your analysis of the probabilities for the late arrival at the Monty Hall game show - provided he did not know which of the remaining two closed boxes had originally been chosen by the contestant.

On the kittens problem, I don't think I am failing to take account of the fact the probabilities can be different for different observers. My point is that as soon as you know at least one of the kittens is male, then the probability, from your perspective, of two males becomes 50%. Where I disagree with Prof Al-Khalili is that I maintain that you don't need to know which kitten is male, for the probability of two males to become 50%.

Just for completeness let's consider the probabilities for 3 different customers of the pet shop - Tom, Dick and Harry.

Tom has only been told that two kittens, a black and a tabby, have been delivered to the pet shop. Tom is not told anything about whether they are male or female.

Dick has been told that one of the kittens has been checked and found to be male. Dick is not told which kitten was checked.

Harry has been told that it is the black kitten that is the male.

So considering the same two kittens in the same pet shop, the probabilities for each of these customers is as follows.

Tom - 25% two males, 25% two females, 25% black male/tabby female, 25% black female/tabby male.

Dick - 50% two males. 25% black male/tabby female, 25% black female/tabby male, 0% two females.

Harry - 50% two males, 50% black male/tabby female, 0% black female/tabby male, 0% two females.

I hadn't heard of Bayesian statistics before, but I looked it up in Wikipedia. Some of it looks pretty complicated - but it does seem to be the appropriate field.
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