A Facebook friend of mine recently challenged his readers to solve this puzzle:
"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"
The "born on a Tuesday" makes for an surprisingly interesting twist, but since I’d like to keep THAT part for a later blog post (in which I hope to convince Dr. Dylan Evans that he is wrong), I’m going to simplify the puzzle to:
"I have two children. One is a boy. What is the probability I have two boys?"
Quite a few people answered: 50% (50-50, 1 in 2).
They were (rightly) informed they were wrong.
Then some people insisted they were right, demanded to be shown how it could NOT be 50% and someone even put a nice amount of money on the answer being 50%.
Unfortunately Dr. Evans didn’t further comment (ignoring the low hanging fruit: that monetary offer), and I posted that I would devote a blog post to it, showing (for free!) WHY the answer is NOT: 50%
The people arguing in favor of 50% are in good company though: their reasoning is also often applied in another famous puzzle: "The Monty Hall problem" that almost everybody gets wrong; even a genius like Paul Erdős initially got this wrong!
I will try to explain what the correct answer is by describing a simple experiment. Even if one doesn’t ‘buy’ or ‘get’ the explanation, simply doing the experiment will be convincing.
For the programmers amongst you, you can also write a simple short program that models the problem: this is how I verified my solution to the Monty Hall problem: it’s VERY convincing (and satisfying) when you see the probability number getting closer and closer to your prediction as the number of iterations over the event increases rapidly!
Before we start: we are allowed to assume that the chance of getting a boy is equal to the chance of getting a girl.
Here we go:
The argument of the "50%" people usually goes something like this:
He has one boy. Therefore the other kid is EITHER a boy, or a girl. The chances of getting either a boy or a girl are the same. There are NO other options. So, how on earth can the chance of getting a kid of certain sex NOT be 1 in 2 (= 50%)?
It’s a fairly intuitive answer, and it’s easy to see why people are even willing to put money on it.
But it’s wrong.
The first thing one needs to understand is this:
The fact that an event has only TWO possible outcomes does NOT necessarily mean that the PROBABILITY of both outcomes are equal!
This is easy to see when I present you with a bowl filled with 1000 ping-pong balls. Each ping-pong ball is either white or black. (if you want to actually DO the following experiment, and don’t have 1000 ping pong balls, use matches, and give the ‘black’ ones a mark with a pen.)
Now I ask you to take (without looking) ONE ball from the bowl.
Note AGAIN that there are ONLY two outcomes here: that ball is either white, or it is black.
So if I ask you what the probability is that the ball you picked is white (this is essentially the same situation as with the kids), would you AGAIN say: "50%"?
You’d be VERY right if you’d answer: well, that depends! It depends on the ratio of white balls to the total number of balls in the bowl: if there are 500 white balls and 500 black balls in the bowl, then yes, the chance of picking a white one is indeed 50%.
But if the bowl contains 999 black balls and only 1 white ball, then your chance of picking that one white ball is NOT 50%! It is a chance of 1 in a 1000, or 0.1%.
In that last case, there are still ONLY two possible outcomes: white or black, but the CHANCE of you picking the white one, while not 0 (meaning: it IS possible), is most definitely not 50%!
A short word to those who would still claim that even in the last case (999 black, 1 white) the chance is still 50%, because: "Hey, there IS a white ball in there, so .. I either grab it, or I don’t: 50-50 chance".
Try applying that to participating in a lottery: Hey, I’ve got a 50% chance of winning the $10,000,000: I either win it, or I don’t, no other possibilities: 50% chance!
We need to keep in mind here what is MEANT by the probability of an outcome of an event. Simply put, the probability of that outcome is the number of occurrences of that outcome when we repeat that event often.
So, while we can then apply that number to a single event’s outcome, (like for instance the chance of winning that lottery, or picking that white ball), that same number is derived from considering what WOULD happen if we replayed our event over and over again. In other words, if you say something has a chance of 50% of happening, then that means, that when you repeat the event often, that ‘something’ happens about 50% of the time, and the more often you repeat the event, the closer to 50% it gets.
Another example: if we throw a die only ONE time, then the PROBABILITY that a given number comes up is NOT 50-50, but 1 out of 6, BECAUSE when we WOULD throw that die MANY times, 1/6 of the times that number would come up.
If you claim that you have a 50% chance of winning the lottery, then what that means (not what you WANT it to mean!) is that if you play that lottery MANY times, you win half of them. Unfortunately, that’s seldom the case with real lotteries, especially the ones with $10,000,000 prizes.
Back to the problem at hand, and let’s rephrase it in terms of a bowl with balls. Since we were allowed to assume that it’s as likely to get a boy as it is to get a girl, we will emulate that ‘equality’ by using equal numbers of white and black balls. Since we also know that probability numbers become more ‘accurate’ with bigger numbers of events, lets put 500 white and 500 black balls in the bowl.
The problem can now be stated as "I grab two balls from the bowl. One is white. What is the chance I got two white balls". The only difference with the ‘kid’ problem is that we can actually PERFORM this experiment.
So let’s do the experiment: grab two balls. there are only three possibilities: 2 whites, 2 blacks, 1 white and 1 black.
Put each pair on their own stack (bb, ww, mixed). Do that for all 1000 balls in the bowl.
So now we have stacks of possible pairs.
All we have to do is count the number of white-white pairs and express that as a ratio of total pairs.
But before we do that, we have to deal with the extra information we were given!
What’s the chance, in the problem scenario, that we picked TWO black balls? That chance is of course ZERO! Because we were told that one of the balls was white!
So we can now simply dispose of the stack of black-black pairs: they don’t play a roll in this problem.
So, now count all the white-white pairs (which is, of course, the entire white-white stack, and none in any other stack), and express them as a percentage of the number of pairs in both the white-white and mixed-color stacks.
That percentage would of course be 50% .. if, and only if, both stacks would be equal in size.
But when you have actually PERFORMED this experiment (and belonged to the "50%" folks), you are now staring at the AMAZING fact that the mixed-color stack is TWICE as large as the white-white stack!
SO, what happened? Where did we go wrong? How did the mixed-color stack end up twice as big as the white-white (and black-black for that matter) stack?
The problem is that for probabilities ‘order’ (as in ‘sequence’) is important.
When counting ‘possible outcomes’ black-white is NOT the same as ‘white-black’, these are two different outcomes. EVEN if that order doesn’t play a role in your problem.
You can see this be grabbing the pairs of balls by using TWO hands: one ball in each. You then make FOUR stacks: black-black, black(in right hand)-white(in left hand), white(in right hand)-black(in left hand), white-white. Another method is to first pick one ball (chance 1/2 it’s white), then pick another ball (change it’s white: also 1/2, total chance 1/2 * 1/2 = 1/4: do that for all combinations, and you’ll get four different combinations, each with a chance of 1/4) and place THOSE pairs in stacks where B-W is a different stack than W-B.
(If you would be using these different possible outcomes for different problems, like, say, forming binary numbers, where white = 0, and black = 1, then it would be easy to see that 01 would be a completely different outcome than 10, as where 00 and 11 would be the same regardless of order of individual bits, so ‘order’ is important for determining possible outcomes).
You will now see that you have FOUR equally sized stacks op pairs of balls. You may then conclude, that 1) the stack B-B can go (since we were given information that one ball was white), and b) that for YOUR problem there’s no meaningful difference between B-W and W-B pairs, so you can lump them together, but then the resulting stack will be twice as big as the original stacks.
(Of course you don’t HAVE to lump them together, you will then have THREE equally sized stacks, with only the white-white stack containing white-white pairs, or, 1/3 of all pairs is white-white)
You will now SEE (even if you still don’t believe or follow the explanation) that 1/3 of all pairs is white-white.
So if you now put all those pairs back in the bowl, you have, of course, a chance of 1/3 to grab a white-white pair. (simply because 1/3 of all pairs in the bowl IS white-white)
And if you’re STILL not convinced: I’d LOVE to play some poker matches with you!
Next time: we’ll add the ‘born on a Tuesday’ clause and have some fun with THAT!
(So, please, no comments on the “Tuesday” part on this post)
I’m always fascinated by the Arizona landscape when we fly over it on our annual trip to Las Vegas.
This year, however, was the first time that I noticed a big crater, a little south of where we were flying, to my feeling not too far from Flagstaff.
I knew there was a big meteor crater in Arizona somewhere and figured this had to be the one.
I just looked it up, and yes, it was the one!
.. learned some interesting facts about it.
I was also intrigued by its official name. Guess what it’s called?
Its official name is:
You may think that this makes sense: calling ‘a’ meteor crater: Meteor Crater.
As it turns out, it is NOT called Meteor Crater, because it is, well, a meteor crater.
Names of natural features are determined by the “US Department of the Interior Division of Names”.
And the “US Department of the Interior Division of Names” has a simple formula for deriving such names: they simply use the name of the post office closest to the natural feature in question.
So, if this post office’s name would have been, say, “Canyon Diablo”, then the crater would now officially have been called “Canyon Diablo Crater”.
Which wouldn’t have been such a bad thing, because that’s exactly what the crater WAS called, before the “US Department of the Interior Division of Names” felt it necessary to change its name to the name of a nearby post office.
And, as it so happened, the name of the post office nearest to this meteor crater was, you guessed it: “Meteor”. Why this post office, located right next to a meteor crater, was called Meteor, is not hard to guess.
So, here we are, with a crater called after a post office, which derived its name from that very same crater.
A bit of a chicked-and-egg situation, although, unlike THAT proverbial problem, we DO know what came first. And even WHEN: some 50,000 years ago, and it wasn’t the post office.
So, what did I learn from all this?
That, apparently, post offices have names.
I bet you have all heard the story that the Mayan Calendar predicts that the world will end on December 12th, 2012. (If not: Google Mayan Calender 2010)
Ha Ha Ha, all those people will feel REALLY stupid when the world actually ends October 21st, 2011!
THAT will teach those 2012 Mayan-End-of-World-ers not to make stupid stuff up!
See: When the World REALLY ends
Now, the Mayan Calendar ends on December 12th, 2012, so if the end of a calendar is to be taken as an indication for this world’s demise, then MY calendar predicts the end of the world to be on December 31st, THIS year!
THAT will teach those people from WeCanKnow not to make up silly stuff!
So, anyway, I’m obviously not buying a 2011 calendars this year!
(Because they’ll be much cheaper in January 2011)
There’s one thing that really puzzles me, though:
The WeCanKnow people are obviously very devout Christians with a very literal view of the Bible. But despite their self confessed claim of a strictly literal interpretation of the Bible, they take a quote from the Bible, and then say: “Err.. NAH, that’s not true”.
Granted, there are some passages in the Bible that are quite nebulous, pure gibberish, actually, so it’s to be expected that people disagree about their meaning.
The mentioned quote however, is not one of those; it’s one of the more straightforward, to-the-point, clear, unambiguous statements. In fact, the WeCanKnow people admit that all the different Christian factions that fight over differences in Biblical interpretations, agree on this one! It’s THAT clear: NOBODY knows when the Rapture will be. NOT EVEN JESUS!
Mark 12:32 (KJV) But of that day and that hour knoweth no man, no, not the angels which are in heaven, neither the Son, but the Father.
Then, given that one holds the Bible as the literal, infallible and inerrant word of an all-knowing and all-powerful Lord, how twisted a mind does one need, to then turn around, point at a VERY clear statement, and say: “Sorry, My Dear Omniscient Lord, but you’re wrong!”?
How can anyone say such a thing without myriads of very loud alarm bells going off and huge red flags being raised in the logic-and-reason department of his or her brain?
The mind boggles.
Oh MY! Have I neglected this blog or what? So much for all my good resolutions.
Let me quickly write something. Let me think. Ah, yes. T-Shirts!
Or rather T-shirt sizes.
There’s something terribly wrong with T-shirt sizes.
T-Shirts come in sizes like S (small), M (medium) and L (large).
These S, M and L sizes remind me of the clothing sizes we got to choose from in the military:
“too small”, “too big” and “WAY too big”.
It’s a known fact that the human species has grown taller over the past millennia, so I must assume that the S, M and L indications go WAY back in time.
Because these days there’s not a single Large person on this earth who would fit into an L sized T-shirt.
So, to allow for the ever growing height of the average human, they had to extend the size range: we now have: XL (Extra large), XXL, XXXL, etc.
I consider myself an average person in height and weight. I’m not tall, nor overly heavy.
My T-shirt size now is XL, and some of those are simply too small!
While it’s kinda silly that an average sized person like myself would need EXTRA EXTRA LARGE T-shirts, that, in itself, is not the problem. It’s just a name.
The REAL problem with T-shirt sizes is: they’re relative!
If you pick a Large T-shirt out of rack, then the L ONLY means that it’s (marginally) larger than an M, from the same brand, make, model and batch.
It does NOT mean that it is of similar size as a different L sized T-shirt.
I now use XL as a baseline size. But I have come across XL T-shirts that I couldn’t even get my head through, as well as ones that probably required me to have a camping permit.
Architecturally, a T-shirt is basically a two dimensional affair: length and width.
However SOME T-shirt makers seem to apply their sizes to only ONE of these dimensions: I have had XL T-shirts that could easily have fit people three times as ‘wide’ as me, yet it wouldn’t cover my belly button.
But I also have had XL T-shirts that were so tight I could barely breath, yet they reached till FAR over my knees.
Are there actually human beings involved in designing these things?
Then there is the type of fabric! While mine are ALL 100% cotton, not all 100% cotton appears to respond equally to washing.
I have had perfect shirts that, after their first laundry experience, would be something like an XXS if that size would exist.
Learning from experience you buy your next T-shirt slightly larger, but IT then emerges from the dryer big enough to cover the entire Statue of Liberty.
One step in the right direction would be to standardize these sizes for T-shirts:
S (M, L, XL, etc) simply means .. a T-shirt is THIS wide, THIS long, the sleeves are THIS wide, and THIS long. Period. Shouldn’t be too difficult: if they can do it for jeans, why not for T-shirts?
Call it a ‘Standard’ T-shirt.
Maybe I should do just that! Establish a set of fixed sizes, trademark them with the name ‘Standard T-shirt’, and sell the rights to use that indication, coupled to certification, to T-shirt makers.
I know ~I~ would only buy ‘Standard T-shirts’ from then on!
Our first day in the New Year was a quiet one. We didn’t go out, other than having our traditional New Year Polar Bear Club meeting (in the pool). For more details see Dawnell’s blog entry.
Other than that, I didn’t do much: played some stupid Facebook game (now that’s something I should have made a New Year’s resolution about!), worked a bit on my picture collection, ignored my other New Year’s resolutions (except for that math thing!) and ordered us a Blue-ray player with Netflix access capability and “Instant Tray Opening” at Amazon I think I will replace our wireless LAN setup with a cat6 wired one, which involves getting up in the attic and making some tricky wall drops.
Yesterday my previous boss (CEO at HF Engineering) asked if I would come along on a trip to a photo store in Orlando (he’s a serious amateur photographer and considering making his hobby his profession). He and I both now do contract work for our previous customer. He told me he’s looking for some cheap office space. I told him that in principle I wouldn’t mind sharing an office space from where we could do our contract work. I like the idea, but it probably means I would have to concentrate on getting more contract work. Plus there is the cost factor. We’ll see.
After that Dawnell and I visited ex-coworker and fellow Dutchie Tim de Waal and his lovely wife Toni and their 9 month old baby Nathan: see Dawnell’s blog entry. They live in a wonderful apartment with a great view over Melbourne’s marina with beautiful sunsets. Unfortunately we had to cut our visit short, as I wasn’t feeing quite well. We’ll do it again soon.
Went to bed early (with a math book!).
This morning my buddy Bob Cuyt called from Belgium (he and his family just got back from his wife’s fatherland Slovenia) and we chatted for a while. He turned 55 yesterday and will become an opa for the third time. Time doesn’t slow down.
Daughter Laurilee and grandson Jackson are on their way up here. Better get dressed.
It’s a somber and cold winter day. 9 C! For Florida, that’s cold! Bitterly cold.
The picture above is my first “Picture of the Day” for 2010 .. Bonnie on her new sofa.
A Wonderful, Interesting, and above all Healthy New Year to all the readers of this blog.
I fully intend to make this year a GREAT year, and I’m NOT going to let me distract from that resolution by the dark, grim, gloomy day it is today.
And on that same note: the cold drizzle is not going to keep me from my traditional "Polar Bear Club" swim either, even though the pool is a MESS right now (The potted conifer, that doubled as Christmas tree this year, just fell in it).
As for New Year’s resolutions: Yes, I made a few. One involves the garage, most others are simply "nonya".
Except maybe the one about my blog: When it comes to blogs (and real life conversations for that matter) I have always been of the opinion that if you don’t have anything to say, well, then don’t!
Until now, that is. At least for my blog.
I intend to start using it more as an actual log book, mostly for myself, just to have a record of my activities, no matter how boring they are.
So expect to find WAY more postings here than in previous years, but don’t expect them to be more than a simple status update for a boring Tuesday afternoon.
As for 2010 …
I’m still unemployed, but have some interesting part-time contract work for, hopefully, another two months.
I will step up my efforts to find full time employment, but at the same time I will be looking for additional contract work.
I’m still trying to figure out what I would prefer: find an interesting full time job, or remain a contract worker, working from home, with the option to make it a business, and grow it into a company with partners and employees. With the economy coming out of a recession, this may be an excellent time to start a company and have it ride the waves up to prosperity.
We’ll be in Vegas for a few days in the 2nd half of January, where Dawnell will try to persuade some slot machines to cover the cost of the trip and where we also will meet some good friends of ours.
We had some plans to spend our vacation traveling the Californian West Coast (well.. duh! California doesn’t HAVE an East Coast), from San Diego to San Francisco.
Just the other day we started to kick around the idea to go to Germany, where Dawnell would love to see the world-famous Oberammergau Passion play, a 10-yearly event. A tradition that started in 1634! It may be followed by a side trip to “The Fatherland” and visit my army of sisters.
I’m sure more plans will be made, before we make our final decision.
Things that I’d like to accomplish but haven’t turned into resolutions to avoid disappointment and self-loathing on 12/31/2010:
- Start (seriously and frequently) practicing and studying the piano
- Start (seriously and frequently) practicing and studying the guitar
- Immerse myself in Linux driver and kernel development
- Start writing apps for Droid phones
- There’s that one subject in math, that I never mastered and would love to master: master it!
- Start reading Feynman’s 3 part “Lectures on Physics” (well ,‘starting’ I already did! Many times!)
- Do the various things around the house that need to be done
- Start writing that best-seller, and have it turn into a blockbuster movie.
- Take more pictures for the “Picture of the Day” club. It’s not called: “Picture of the Week” club!
- Well, here I CAN reveal some – but not all – resolutions:
- go to a fitness club and start working out, even it it’s only once a week, or 10 minutes at a time.
- two words: medication: discipline.
And with that …
ON to the pool!
I was just informed about my previous employer’s (Conexant) sudden stock price hike.
Checked the news and read about the sale of their broadband access division to Ikanos.
Ikanos is headed by CEO Michael Gulett.
Who once was the president and COO of Virata Corporation!
Sounds like he’s buying back his own stuff, from before his disastrous ‘merge’ with Globespan, the company that then went on to buy Intersil’s (also a previous employer of mine) Wireless Division (Harris’s spun-off semiconductor division), and then later ‘merged’ *cough* with Conexant.
So now Conexant is selling the stuff back to the ex-Virata president’s company! Ha!
Dang, there’s a lot of hot potato-ing going on in the semiconductor industry, these days (years!).
Here’s another quiz:
Consider a list of naturally occurring numbers. And by naturally occurring is meant numbers like for example stock prices, number of inhabitants of cities, your electricity bills of the past few years, prices on your Saturday grocery receipt, lengths of rivers, number of books in the bookcases of all your friends, you name it.
The question then is about how often a particular digit appears as the first digit of the numbers in such a list.
Let’s take for example the digit ‘1’. The question then becomes: how often (in %) will ‘1’ be the first digit of these numbers in a given list? Or to put the same question differently: How many numbers in that list (in % of the total number of numbers) start with a ’1′?
Hint: keep in mind the first category this post is published in!