Page 1 of 2

A debate over some simple probabilities

Posted: Fri Jun 01, 2007 11:12 pm UTC
by hencethus
I happened to visit the same restaurant twice in the same week, accompanied by different people each time. Both times it was someone else (besides me) that chose where we sat. And both times we happened to sit in the same place.

On the second visit I mentioned that we were sitting in the same place that I'd sat the last time I'd been there, and someone else asked (probably rhetorically), "What are the odds of that?"* So I said that assuming there were 50 places to sit, and assuming that our seating selection was truly random with an equal chance of sitting at any of the 50 places (which of course isn't true), then the odds were 1 in 2500.

Immediately everyone protested. My companions insisted that the odds were in fact 1 in 50 since the initial seating choice was "a given" (whatever that means). One of my friends even used an analogy, saying that by my logic, if he flipped a coin I would have a 1 in 4 chance of guessing correctly on which side it would land.

I tried to explain that while the odds of choosing correctly were 1 in 2, the odds of him flipping heads and me randomly guessing heads specifically was only 1 in 4 (and that the same was true for tails).

Another one of my companions again insisted that I was wrong, and suggested that I go back to school and take a statistics course.

So who's right here, guys?

*Addendum: We're talking about the odds of sitting in that specific place twice, which I thought was made clear by my explanation of the coin analogy, but apparently I'm biased. :P

Posted: Fri Jun 01, 2007 11:15 pm UTC
by Mighty Jalapeno
You're right, and they're probably from Texas.

Posted: Fri Jun 01, 2007 11:27 pm UTC
by Woxor
It depends on what the exact question is, but I'm afraid I think you're wrong.

If the question were, "What are the odds of sitting in THIS place twice in a row?" you'd be right. But as you've phrased it, they asked the odds of sitting in "the same place as last time" the second time you come. The odds of the latter happening are indeed 1 in 50, since there are 2500 possible ways to be seated twice, and 50 possible ways to be seated at the same place twice.

But what do I know, I'm from Texas. Maybe math is different down there.

Posted: Fri Jun 01, 2007 11:32 pm UTC
by Vaniver
You're wrong.

The probability is sitting in the same spot.

The specific place the first spot is in doesn't matter- just if the second is the same. Just like it doesn't matter whether you guessed heads or tails; just whether or not you were right. 1/4 (guessed heads and it was heads) + 1/4 (guessed tails and it was tails) = 1/2 (guessed what it was); you can do a similar addition for each table, or just accept the principle that the first is a "given", and you're just comparing if they are the same.

[edit]I grew up in Texas as well. Odd.

Posted: Fri Jun 01, 2007 11:33 pm UTC
by hencethus
Woxor wrote:If the question were, "What are the odds of sitting in THIS place twice in a row?" you'd be right.


That is indeed the question. Otherwise I wouldn't have bothered with the coin flipping explanation.

Posted: Fri Jun 01, 2007 11:35 pm UTC
by Woxor
hencethus wrote:That is indeed the question. Otherwise I wouldn't have bothered with the coin flipping explanation.

Then you should probably change this:
hencethus wrote:On the second visit I mentioned that we were sitting in the same place that I'd sat the last time I'd been there, and someone else asked (probably rhetorically), "What are the odds of that?"

... because that's not the same question.

Posted: Fri Jun 01, 2007 11:38 pm UTC
by Vaniver
As well, the question "will an event be repeated twice" is generally more frequently asked than "will this event be repeated twice," unless there is something to differentiate "this event" from other events. If it's just the coincidence of sitting at the table twice, the first is what people will expect. If it's the coincidence of being seated by the kitchen, or bathroom, or wherever, twice, they might expect the second.

Posted: Fri Jun 01, 2007 11:43 pm UTC
by hencethus
Woxor wrote:... that's not the same question.

Isn't it? I think it's phrased rather ambiguously. Anyhow, although I didn't include it in my post, I did explain to my friends that I was talking about the odds of sitting in that specific spot both times. They still insisted that the original seating selection was a given regardless.

I'll make an addendum to the original post, though.

Posted: Fri Jun 01, 2007 11:45 pm UTC
by Woxor
Yeah, I mean it would be pretty silly if someone asked, "What are the odds that I will get heads when I flip a coin?" and you replied with "About 1 in a trillion because there are a billion places to stand and 500 things you could do, so the chance of you standing in that exact spot and flipping a coin to begin with are about 1 in 500 billion." They weren't asking what the probability of the universe being in a given state is, they were asking what the chances that a coincidence like that would occur.

EDIT:
hencethus wrote:I did explain to my friends that I was talking about the odds of sitting in that specific spot both times. They still insisted that the original seating selection was a given regardless.

Okay, if they still disagreed with you, then you bested them mathematically, but I agree with Vaniver's post. vvvvvvvvv

Posted: Fri Jun 01, 2007 11:46 pm UTC
by Vaniver
I did explain to my friends that I was talking about the odds of sitting in that specific spot both times. They still insisted that the original seating selection was a given regardless.
Well, that still seems ambiguous. I mean, unless it's special for a reason other than you sitting there, asking about the odds of sitting at table 16 twice in a row is a silly question. Since the odds are high you would have commented on a repeated seating regardless of where you sat, your friends probably interpreted the question the way we did, regardless of you rephrasing it to a less meaningful question.

Posted: Fri Jun 01, 2007 11:52 pm UTC
by hencethus
Vaniver wrote:Well, that still seems ambiguous. I mean, unless it's special for a reason other than you sitting there, asking about the odds of sitting at table 16 twice in a row is a silly question. Since the odds are high you would have commented on a repeated seating regardless of where you sat, your friends probably interpreted the question the way we did, regardless of you rephrasing it to a less meaningful question.

Agreed. Eventually I conceded that we were both right depending on how the problem was approached, but they made no such concession. The initial seating was "a given." Period.

Posted: Fri Jun 01, 2007 11:57 pm UTC
by Vaniver
The initial seating was "a given." Period.
Well, was there anything special about the table, other than that you sat there?

You're starting the thought process from before the first time you went to the restaurant, when you should be starting it the second time you enter the restaurant.

Posted: Sat Jun 02, 2007 12:03 am UTC
by hencethus
Vaniver wrote:Well, was there anything special about the table, other than that you sat there?

It was right by the trash bins, which I why I remembered that it was the same table in the first place. I hate sitting by the trash.

Vaniver wrote:You're starting the thought process from before the first time you went to the restaurant, when you should be starting it the second time you enter the restaurant.

I don't think there's a time from which I should start my thought process. Just as there's no reason that I shouldn't ask the question, "If we both flip a coin now, what are the odds that both coins will land on heads?" The answer is 1 in 4. It would be silly to say, "No, you shouldn't ask that question. The answer is 1 in 2 because the coins could also both land on tails."

Posted: Sat Jun 02, 2007 1:19 am UTC
by Woxor
hencethus wrote:I don't think there's a time from which I should start my thought process. Just as there's no reason that I shouldn't ask the question, "If we both flip a coin now, what are the odds that both coins will land on heads?" The answer is 1 in 4. It would be silly to say, "No, you shouldn't ask that question. The answer is 1 in 2 because the coins could also both land on tails."

I think it makes sense to say what the natural question to consider is, though. Ex: my 1-in-a-trillion coin-flipping example above.

Posted: Sat Jun 02, 2007 1:21 am UTC
by 3.14159265...
"What are the odds of that?"


Well it was surprising to you that you sat at the same table twice not that you sat at table (49 or w.e.) twice.

So they are right.

No concessions!

Posted: Sat Jun 02, 2007 1:34 am UTC
by hencethus
Woxor wrote:I think it makes sense to say what the natural question to consider is, though. Ex: my 1-in-a-trillion coin-flipping example above.

Natural to whom? Anyway, although I understand the point you're making, I don't think your reductio ad absurdum is especially relevant here. Obviously the question was quite natural to me.

3.14159265... wrote:Well it was surprising to you that you sat at the same table twice not that you sat at table (49 or w.e.) twice.

It was surprising that "we were sitting in the same place that I'd sat the last time" (exactly how it was phrased in the original post), which was a very specific place (table 49 or whatever).

Posted: Sat Jun 02, 2007 1:35 am UTC
by The_Spectre
I'll tell you a shocking story. This morning, I parked my bike next to a car that had the following license plate number:

JF-24-UE

Amazing, isn't it? I mean, the odds of that exact license plate number are really really tiny!

Posted: Sat Jun 02, 2007 1:47 am UTC
by hencethus
The_Spectre wrote:I'll tell you a shocking story. This morning, I parked my bike next to a car that had the following license plate number:

JF-24-UE

Amazing, isn't it? I mean, the odds of that exact license plate number are really really tiny!

Even more shocking: I was playing a game of hearts, and I was dealt 13 random cards: J(c) 3(d) K(d) A(d) 2(s) 8(s) J(s) Q(s) K(s) A(s) 3(h) 10(h) Q(h).

The odds of being dealt that exact hand are something like 1 in 600 billion. Amazing! I can hardly believe it happened!

Posted: Sat Jun 02, 2007 2:52 am UTC
by Darth Eru
I agree with the people saying that it was understandable your friends misunderstood the question. The first time you sat at that particular table, it didn't really matter which table you sat at. It only becomes significant once you sit at the same table later on, so the significant question is what the odds are of sitting at any table twice, not a particular table twice.

Consider this: The chances of sitting at a particular table twice are the same as those of sitting at any two particular tables, whereas the chances of sitting at the a table twice are very small compared to the chance of sitting at two different tables.

Posted: Sat Jun 02, 2007 2:59 am UTC
by bitwiseshiftleft
A mathematician is flying from America to Europe, and when his bag is searched at security, the screener finds a bomb in it! He angrily asks the mathematician for an explanation. "Well," the mathematician explains, "I wasn't going to set it off, and what are the odds there being two bombs on the same flight?"

Posted: Sat Jun 02, 2007 3:05 am UTC
by Woxor
hencethus wrote:It was surprising that "we were sitting in the same place that I'd sat the last time" (exactly how it was phrased in the original post), which was a very specific place (table 49 or whatever).

This clarification only makes it worse for your case. The chances of "sitting in the same place that you sat last time" are 1 in 50.

hencethus wrote:Natural to whom? Anyway, although I understand the point you're making, I don't think your reductio ad absurdum is especially relevant here. Obviously the question was quite natural to me.

The reason it was more natural to ask the 1 in 50 question is that the past event had already happened, and the second seating provoked the question. Like someone else said above, it's the fact that you sat in the same place that was of interest, not that you sat at table "49" in particular twice. You're right about your probability, but I think it would have been more reasonable to interpret the question in the 1 in 50 sense.

And my 1 in a trillion example makes a valid point: it's important to consider what part of the problem is relevant. The penny flipping is what we care about, not where you are or how you do it. Similarly, the coincidence of sitting at the same table is what we care about, not which table it was.

Posted: Sat Jun 02, 2007 5:35 am UTC
by 3.14159265...
It was surprising that "we were sitting in the same place that I'd sat the last time" (exactly how it was phrased in the original post), which was a very specific place (table 49 or whatever).


Would you have been just as surprised if you sat at table 48 or 47 twice, if yes they are right.

If that table is special in that it was a bad one, or always served by the hot waitress or something else then you are.

I am sure its the first case, so you are wrong, concede 100%! now!

Posted: Sat Jun 02, 2007 5:38 am UTC
by Woxor
3.14159265... wrote:If that table is special in that it was a bad one, or always served by the hot waitress or something else then you are.

He said it was by the trash, so he's got a leg to stand on. I just think his friends had two legs. :D

Posted: Sat Jun 02, 2007 8:53 am UTC
by Zohar
I think you got a good enough answer already (if you asked what are the chances to pick this specific table twice, it's 1/2500, if it's to sit twice at the same table it's 1/50).

I just thought to mention that everyone assumes a uniform distribution on the tables (not that it matters for the sake of the argument). Obviously they aren't. Some people have favorite tables, some don't want to sit in the center of the room, the good areas might get crowded and noisy so you'd choose a "worse" table etc. I guess that would be interesting to see. The different distributions according to different people, kind of like the choose the right urinal test.

Posted: Sat Jun 02, 2007 11:26 pm UTC
by Yakk
So you walk into the restaurant, and your friend chooses a table.

You sit down and say "man, we came into the restaurant, and we sat at the table my friend chose! What are the odds of that? You know, assuming we sit where my friend chooses."

The transformation of the situation, where you change the table on both visit 1 and visit 2 to the same different table, generates a situation where saying "what are the odds we sit at the same table twice?" is equally valid.

When one talks about how unlikely something is, you should talk about how unlikely it is up to such automorphisms, and up to "even more unlikely things" (which is a hard part).

"What are the odds I would have to sit next to the trash two visits in a row?"
vs
"What are odds we sit in the same spot two days in a row?"

The first one is 1/2500 (assuming only one set of seats next to the trash)
The second one is 1/50.

Posted: Sat Jun 02, 2007 11:31 pm UTC
by __Kit
I agree with your friends sorry, I have often pondered this. If I roll a dice it doesn't matter WHAT it is (1-6, lets say 3) th chance of me getting the same number as the first time is 1 in 6

Posted: Mon Jun 04, 2007 2:39 am UTC
by Blatm
To repeat what's been said about a dozen times, it all depends on what you mean. Your friends are right on a practical level, but you're right on a technical level, and being technicaly correct is the best kind of correct. I voted in your favour, but only because the poll isn't going to matter. If it were, I would probably vote for you both being right. Actualy, I'm very surprised that only one person has picked option 4 up to now.

Have fun with your pointless bickering. :)

Posted: Mon Jun 04, 2007 2:56 am UTC
by Woxor
Blatm wrote:Your friends are right on a practical level, but you're right on a technical level, and being technicaly correct is the best kind of correct.

"Dammit hencethus! They've got us surrounded! I thought you said the odds of them bombing Pearl Harbor twice were one in 2500, over!"

"Well, there are 50 U.S. states, so 50 squared is 2500, over!"

"But we already knew that Pearl Harbor was bombed once! Oh God, I'm going down! Mayday!"

"You didn't specify what the exact question-" *BOOM*

Posted: Mon Jun 04, 2007 7:15 am UTC
by Gelsamel
Jesus fucking christ, if you explained that you're talking about on both occasions and they still insisted it was 1 in 50 then they're retarded.

The chance of getting 2 heads in a row is 1/4 do a fucking probability tree.

Code: Select all

        Heads 1/4
 Heads<
/       Tails 1/4
\       Heads 1/4
 Tails<
        Tails 1/4






However the probability of getting the same face of the coin as last time is 1/2.

Code: Select all

        Heads 1/2
 Heads<
        Tails 1/2




If you made it clear you were talking about the first situation and they dispute that then they're idiots. And There is no use in trying to reason with them.

Posted: Mon Jun 04, 2007 7:43 am UTC
by scowdich
Woxor wrote:"Dammit hencethus! They've got us surrounded! I thought you said the odds of them bombing Pearl Harbor twice were one in 2500, over!"

"Well, there are 50 U.S. states, so 50 squared is 2500, over!"

"But we already knew that Pearl Harbor was bombed once! Oh God, I'm going down! Mayday!"

"You didn't specify what the exact question-" *BOOM*


For the record, there were only 48 U.S. states when Pearl Harbor was bombed (the first time), resulting in a "technical probability" of 1/2400, not 1/2500. Or perhaps since Hawaii wasn't a state then, using a base probability of 1/48 won't work...hmm.

Posted: Mon Jun 04, 2007 10:40 am UTC
by evilbeanfiend
Gelsamel wrote:Jesus fucking christ, if you explained that you're talking about on both occasions and they still insisted it was 1 in 50 then they're retarded.

The chance of getting 2 heads in a row is 1/4 do a fucking probability tree.

Code: Select all

        Heads 1/4
 Heads<
/       Tails 1/4
\       Heads 1/4
 Tails<
        Tails 1/4





i think their point is that the first toss has already happened so we know exactly how it came out .'. assigning a probability other than 1 to this event is meaningless.

Posted: Mon Jun 04, 2007 11:13 am UTC
by Gelsamel
evilbeanfiend wrote:i think their point is that the first toss has already happened so we know exactly how it came out .'. assigning a probability other than 1 to this event is meaningless.


No, "We know how the first decision came out therefore the answer is 1/2" is the 2nd tree I came up with. Note how Heads starts the chain, because it already happened.

The OP said he explained that he was evaluating the hypothetical "What are the chances?" and they still disagreed.

And the hypothetical isn't meaningless, otherwise we could answer questions like "Pr(2 heads in a row) = 1/2 because after the first I'll know the side"

Or why not just flip 2 coins and if it's Not the same Say Pr=0, or if it is say Pr=1?

Or http://xkcd.com/c221.html

To the question "What is the chance of sitting in the same seat twice in a row" IS actually 1/50. But the question "What is the chance of sitting in this particular seat twice is a row" is 1/2500.

ie.


"What is the chance of getting the same side twice in a row."

Code: Select all

        Heads 1/4 <------
 Heads<
/       Tails 1/4  AND
\       Heads 1/4
 Tails<
        Tails 1/4 <------

= 1/2



But "What is the chance of getting this particular side twice in a row"

Code: Select all

        Heads 1/4 <------
 Heads<
/       Tails 1/4    OR
\       Heads 1/4
 Tails<
        Tails 1/4  <-------

=1/4



The OP was answering the 2nd.

Sure there was obviously a heap of ambiguity which explains why they disagreed initially (really they should've realised this ambiguity and asked the OP to clarify).

But to continue to disagree after the OP clarified is ridiculous. Unless you disagree with the method of mathematics altogether in which case you better have a damn good reason.

Posted: Mon Jun 04, 2007 12:26 pm UTC
by evilbeanfiend
yes but i think they are trying to point out that the hypothetical question answered by the op is always hypothetical
at the time the question was asked because 1 outcome is known.

i.e we have these possible questions

q1. given 2d50 what is the chance of getting 50 on each before we roll
q2. given 2d50 what is the chance of getting the same on each before we roll
q3. given 2d50, and the first die has already been thrown and is known to be 50, what is the chance of getting 50 on both.

i think they are interpreting it as q3, and given that the other two are purely theoretical questions as they don't considered the fact that one roll has already been observed .'. i think their position that they are right and the op is wrong is defensible even though i think the op being right is also a defensible position. i.e. it doesn't just depend on what the question was but when it was asked

Posted: Mon Jun 04, 2007 1:12 pm UTC
by Gelsamel
Right, they were interpreting it differently, yet when the OP explained what he was answering they still disagreed.

And I wouldn't give them the benefit of the doubt saying that they were 'arguing the OP's question is always hypothetical' all the OP has said is that they disagreed with his calculation of HIS question (Question 1 in your list).

And the OP's question isn't always hypothetical: "If we choose one of the 50 seats totally at random then within the next two times we go to the restaurant what is the probability that we end up at the same seat both times?"

The fact of the matter is Question 3 is NOT useful, it tells us NOTHING.

Why? It sets any events that has happened = 1. And of course that's true, but does it tell us anything? No. "What is the chance that we got the exact same seat as last time?" "100%, since we did get it again"

"Wow I won the lotto!"
"Wow, what're the odds of that?!"
"100% chance, since I did win it!"
Of course once again in hindsight this is perfectly correct, however it tells you shit all.

The only meaningful questions are 1 and 2 in your list because they actually tell you something.

So unless someone is looking for a totally meaningless answer when they ask "What is the chance of that?" Or "What are the odds?", then the question is either 1 or 2, and which one of those two it is depends on the situation.

Posted: Mon Jun 04, 2007 2:00 pm UTC
by evilbeanfiend
Gelsamel wrote:The fact of the matter is Question 3 is NOT useful, it tells us NOTHING.


it does tell us something, it tells us the probability of one roll and makes it explicit that we consider previous rolls to have no effect on future rolls, this is a point that is often missed by the general public (although pretty obvious to a bunch of mathematicians) it is important however as you clearly get a different (and very wrong result) if you modify your expectation of future rolls depending on the value of previous rolls (this all depends on the events being genuinely random of course, if they are highly correlated then modifying your expectation depending on past results is exactly the right thing to do)

Posted: Mon Jun 04, 2007 2:08 pm UTC
by Andrew
I went out for a meal one day, and I sat in a different place to last time. The odds of me sitting in those two places in that order must be 1 in 2500. Big whoop. Yeah, so a 1 in 2500 chance came up. But there were 2500 mutually exclusive one-in-2500 chances. One of them had to come up. It's highly unlikely, but it's not special. There were 50 possible combinations of chairs that would have caused you to comment in this way, and you had a 1 in 50 chance of hitting one.

It's all well and good saying "they're answering the wrong question" but this is a conversation, not an arithmetic exam. It's entirely reasonable for them to tell you you're asking the wrong question.

Posted: Mon Jun 04, 2007 2:13 pm UTC
by Scarblac
I like the dice analogy.

If you roll a 5, and then you roll a 5 again, then say "Gee, the same number as last time! What are the odds of that!" then the answer is 1 in 6, period.

If you then try to argue that you meant that specific number, not just any same number, so the answer should be 1 in 36 - then that just makes you look lame, since that obviously wasn't what you meant initially.

Posted: Mon Jun 04, 2007 2:17 pm UTC
by Gelsamel
evilbeanfiend wrote:
Gelsamel wrote:The fact of the matter is Question 3 is NOT useful, it tells us NOTHING.


it does tell us something, it tells us the probability of one roll and makes it explicit that we consider previous rolls to have no effect on future rolls, this is a point that is often missed by the general public (although pretty obvious to a bunch of mathematicians) it is important however as you clearly get a different (and very wrong result) if you modify your expectation of future rolls depending on the value of previous rolls (this all depends on the events being genuinely random of course, if they are highly correlated then modifying your expectation depending on past results is exactly the right thing to do)


.......We're not "modify[ing] your expectation of future rolls depending on the value of previous rolls".


It's not as if we're saying the chance of picking 1 of 50 seats randomly is 1/2500.

We're saying the chance that seat 50 is chosen 2 times in a row totally randomly out of 50 seats is 1/2500, there is NOTHING wrong with that.

Random Choice 1 in question 1 and 2 has NO EFFECT on Random Choice 2, no effect in this situation AT ALL. However the WHOLE probability tree of TWO choices gives you a 1/2500 chance, 1/50 each time.

What I meant is by setting all happened events = 1 you LOSE information. And I was also saying since they asked the question after they HAD sat down at the same table wouldn't that happened event = 1?

Setting events = 1 is useless. It is better to totally IGNORE them, in which case 1d50 suffices. Of course then you're not answering the question, which is why only question 1 or 2 can answer "What are the odds?" meaningfully.

Posted: Mon Jun 04, 2007 2:37 pm UTC
by Woxor
Gelsamel wrote:.......We're not "modify[ing] your expectation of future rolls depending on the value of previous rolls".


It's not as if we're saying the chance of picking 1 of 50 seats randomly is 1/2500.

We're saying the chance that seat 50 is chosen 2 times in a row totally randomly out of 50 seats is 1/2500, there is NOTHING wrong with that.

Random Choice 1 in question 1 and 2 has NO EFFECT on Random Choice 2, no effect in this situation AT ALL. However the WHOLE probability tree of TWO choices gives you a 1/2500 chance, 1/50 each time.

No one's disputing the mathematics. We're disputing the relevance.

Gelsamel wrote:What I meant is by setting all happened events = 1 you LOSE information. And I was also saying since they asked the question after they HAD sat down at the same table wouldn't that happened event = 1?

No, you gain/retain the information that it happened. You can figure out what the prior probability was, and no, there's nothing wrong with doing that, you just shouldn't pretend it's relevant when it's not.

Gelsamel wrote:Setting events = 1 is useless. It is better to totally IGNORE them, in which case 1d50 suffices. Of course then you're not answering the question, which is why only question 1 or 2 can answer "What are the odds?" meaningfully.

This is just nonsense. Ignoring past events isn't better in any way; interpreting them correctly is. So long as your epistemology is certain, events that have already happened have probability 1, but more importantly, they did happen (since events can have probability 1 and still not happen). This allows us to derive probabilities of other things that do depend on them with greater accuracy.

As many people in this thread have said a dozen different ways, the fact that it was in the given place twice was not what made the event notable and is thus irrelevant to the natural question it raised. The OP is just trying to cover for being bad at interpreting their question by hiding behind an irrelevant question that he WAS right about. I can't understand why you agree with him, though.

Questions 1 and 2 were only relevant before the event transpired, and Question 1 was still only relevant if the place in question was special in some way. These weren't the conditions when the question was asked, so they are irrelevant.

Posted: Mon Jun 04, 2007 2:47 pm UTC
by evilbeanfiend
Gelsamel wrote:.......We're not "modify[ing] your expectation of future rolls depending on the value of previous rolls".


no, i know that, that's cos you are answering a different question. they are perfectly allowed to think that your question is irrelevant, in that way their position is defensible (note i'm not saying correct).

glossing over ignoring previous rolls is all very well but so many people make mistakes about that (e.g. "number 43 has come up every 3 weeks in a row in the lottery, so it must be less likely it will come up now!" except it isn't) so i'm not surprised the op's friends seem to be concerned that this sort of mistake is being made (even though it isn't) hence their instance that the first table was 'a given'.