# Intuition, and the Monty Hall Problem

Statistics is hard. It’s one of those fields, where your intuition seems to take you gently and seductively towards safety, and then you wake up in a bathtub full of ice, missing a kidney. Interface design is often the same way (but, for once, that’s not what I’m writing about). When we say that we find something “intuitive” it means that we have seen something like it before—we are saying that it is familiar. The thing with statistics is that similar looking situations can behave very differently.

The first time I had this point driven home was when I was presented the Monty Hall problem. Posit: You are on a game show and three doors stand before you. Behind two of the doors lies nothing, behind one door is a pile of cash. The game begins by you picking one of the doors. The host (Monty Hall) then opens one of the two remaining doors, revealing one that is empty. Then it’s your choice: Do you stick with the door you originally picked, or do you switch to the other door? Does it matter?

Intuition says is that it doesn’t matter. It’s 50%-50%. It’s the same as throwing a coin—the money is in either one door or the other. This kind of thinking will lose you a kidney. You’ve been tricked by “intuition” to not got suckered by the gamblers fallacy, only to get suckered by something new.

It’s best to switch doors. In fact, switching doors will make you win twice as often as staying. While I’ve been able to prove that for a long time—by enumerating all possible outcomes and counting&mash;it never really clicked in my head *why* it’s true. It never became familiar. It never became intuitive. Until a couple days ago. Here’s the explanation the occurred to me (in the shower, as always) that made it all make sense:

The chance that you pick the right door on the first choice is 1/3: there are empties and one prize. Conversely, that means that there is a 2/3 probability that prize is in *one* of the other two doors.

If you could open the other two doors, see which ones contains the money, and take it, you would have clearly do that. That’s twice as good as getting to open only one door! Well, when Monty opens one of those doors that doesn’t contain the prize, he’s effectively giving you that ability: there is still a 2/3 probability that the prize is behind the set of doors you didn’t pick.

It’s obvious, now, that switching will give you a 2/3 chance of winning. By opening a door that doesn’t contain a prize, Monty has effectively concentrated that 2/3rds probability into the one remaining door.

This way of looking at it makes it really easy to calculate similar problems. For example, take the 5 door case: you pick a door, Monty opens 3 empty doors, and you have to choose if you should switch. Your first pick has a 1/5 chance of getting the prize, which means that there is a 4/5 chance that it is in the set of doors you didn’t pick. Monty opens 3 of those 4 doors, which means that 4th unopened door takes that full 4/5 probability. Thus, switching is 4 times more likely to win than is staying.

What happens if Monty only opens two doors, instead of three? Should you switch? It’s easy to figure out. You still have 1/5 chance of getting the door on your first pick. The other doors still have a 4/5 chance of having the prize. He opens two, which means that 4/5 probability is spread equally over the remaining two doors. Thus switching will give you a probability of half of 4/5ths, which is a 4/10ths probability of winning. That’s better than 1/5 chance of winning, so switching is better.

What’s the moral of the story? That intuition is malleable and that statistics is hard, until you find the right way of framing the problem. And that’s a general rule.

RT @aza Intuition, and the Monty Hall Problem | Follow @aza on Twitter | All blog posts

No related posts.

Tags: intuition, monty hall problem, statistic

Dan

Huh? Once I got to your second picture the little mathematician inside me cried out and I just had to post. It’s been a while since I took Probabilities and Statistics but this is simple enough compared to some of the stuff I had to do in that class.

Once Monty opens the second door, you are given new information (that the door does NOT contain the money). Now your probabilities change to 1/2 for each of the still unopened doors… it’s still 50/50. Not 33.3/66.7.

2/3 only works if you didn’t know what was behind the door Monty picked and you could pick to open BOTH at once.

The same applies to your first 5 door example. The chance is still 50/50. because the three opened doors now have 0 probability (since there’s no way the money is there since Monty just showed you it isn’t) which redistributes the full 100% probability equally among the other two doors.

For your last five door example, two doors are opened and their probability drops to 0, as before. Now you have three doors, and since you have no information about which door is more likely to contain the money (assuming the assignment of the money to a door is completely random every time) every door has a 1/3 probability.

Going back to your original reasoning for your method, you claim you enumerated all possible outcomes for the problem and found your 66.7% success rate with your strategy. OK I’m gonna see if I can figure out what you’re thinking here.

We have three doors, times three choices you can make. Depending on the door and the first choice Monty has either one or two empty doors to choose from to show you. Then you have your second choice of one of two doors. Choosing to switch is 50% of this sample, while staying is the other 50%.

I’ll number the doors 1, 2, and 3, and tally the results below in a little table with columns: Money Door, First Choice, Monty Door, Second Choice, and Won

Stay:

1 1 2 1 Y

1 1 3 1 Y

1 2 3 2 N

1 3 2 3 N

2 1 3 1 N

2 2 1 2 Y

2 2 3 2 Y

2 3 1 3 N

3 1 2 1 N

3 2 1 2 N

3 3 1 3 Y

3 3 2 3 Y

Switch:

1 1 2 3 N

1 1 3 2 N

1 2 3 1 Y

1 3 2 1 Y

2 1 3 2 Y

2 2 1 3 N

2 2 3 1 N

2 3 1 2 Y

3 1 2 3 Y

3 2 1 3 Y

3 3 1 2 N

3 3 2 1 N

6/6 and 6/6.. The only way I see you can get a better chance than 50% could be different is if whoever hides the money or Monty uses a pattern instead of complete randomness and you pick up on it, then it gets interesting (if you realize Monty always picks the smallest numbered empty door, for instance, whenever he picks 2 you know for sure the money is behind door #1). But this has nothing to do with your strategy.

I think you have succeeded in showing statistics is hard, just not the way you were trying. :) Sorry. I honestly have no clue how you’re coming up with your probabilities.

Dan

After further thought, the only way I see this working is if you make the first choice win the player money if they get it right… but then if it’s wrong (2/3 chance) monty will either pick your door or the other empty door. If he picks yours you have a 50/50 chance, if the other one your chance goes up to 100% since there’s only one unknown door left. So the chance works out to 1/3 + (2/3 * 1/2) = 2/3

But if that’s it, “staying” would be ludicrous since you already know your first guess was wrong, and there’d be no reason NOT to switch, and whoever is putting the money behind doors is swamped with Statistics students who win all his money and he goes bankrupt. Unless he went to Statistics class too and charges enough money to offset the 2/3 probability (IE at least 2/3 the prize).

Wikipedia mentions the problem you’re talking about here so I guess there must be something to it, but I’m still not seeing it.

Dan

OK, found the full Wiki article, I’m gonna go figure this out now. :)

Mikael

So if there is a hundred doors and he opens 98 of the empty ones, that means you will have a 1% chance that your first choice is right and if you take the other door a 98% chance of getting rich?

Yes but only if your first choice was not one of the two doors still left unopened, if the first choice was one of the unopened there would be a 98% probability on both.

And that means it is still a 50%/50% (98/98) chance on getting the gold.

So switching door will not give you a better chance or probability or anything else for that matter.

Just my two cents.

Axel Hecht

Aza’s right, up to the point that I don’t think this is actually statistics, but combinatorics.

Good old days in school, I don’t yet know whether our teacher didn’t get it himself, but that topic was hell confusing up to the point where I actually discarded what he said and picked up the book over the weekend.

One detail that’s been missing in the comments here is that Monty *never* opens the door you picked.

The real trick here is to model the additional knowledge that you get from Monty opening one door. And the best way to do that is go back to the original set like Aza did. Assuming that you’re standing in front of a problem that you don’t know anything about (i.e. just two doors) is wrong, so even if you have two doors, the probability is really not 50 50.

As you can tell from the chances, if you’re actually in the show and only have one go through the game, you still have a one third chance of getting nothing.

[ICR]

This in conjunction with the Wikipedia explination helped me to understand this (http://en.wikipedia.org/wiki/Monty_hall_problem).

Here’s how I understand it, the same as Aza just phrased slightly differently:

The probability that, in the first instance, you picked the right door is 1/3. Thus the probability that you picked the wrong door is 2/3 – the car is twice as likely to be behind one of the other doors. Even after one of the duff doors has been opened, the probability of having picked the wrong door in the first instance is still 2/3. It’s just now, as Aza says, concentrated on the one remaining other door.

[ICR]

Okay, according to Wikipedia that’s wrong. It works in this instance, but not to more generalised problems. Bugger.

Ron

@Dan, Mikael: Comfy in that bathtub of ice? You’d thnk when someone says “hey, this is hard and doesn’t work the way you’d think”, you wouldn’t be quite so quick to jump up and make total asses of yourselves. And Mikael, you might want to switch from cents to pesos for future posts–it’ll be a more accurate valuation.

[ICR]

Oh no, it does still work (I think) :S I’ll stop posting now.

Dorus

I do not completely agree with this explanation, even when it’s a good way to visualize the problem, it’s not complete. Wikipedia explain it right;

Crucial about the door problem is that:

1. the game master never opens the door you picked.

2. the game master never opens the door with the money.

The first fact is explicit named in the initial question, so no need to assume anything about that. The second fact is a bit more tricky to find. If we do allow the game master to open the door with the money, the game master does not add any new information to the game, he just eliminate a random door, what mean that the remaining 2 doors keep 1/2 change to be correct, but also that the change the game is a fluke is 1/3th.

Sander

For the skeptics, here is a simulation you can run in Python:

from random import randint

games = switch_win = not_switch_win = 0

for games in range(0, 100000):

prize = randint(1, 3)

first_choice = randint(1, 3)

if prize is first_choice:

shown = randint(1, 2)

if shown is first_choice:

shown += 1

else:

shown = 6 – prize – first_choice

if first_choice is prize:

not_switch_win += 1

else:

switch_win += 1

print ‘games: %d’ % (games + 1)

print ‘won by switching: %d’ % switch_win

print ‘won by staying: %d’ % not_switch_win

The results here looked like:

games: 100000

won by switching: 66729

won by staying: 33271

Sander

Hm, wait, logics seem to have escaped me for this morning. The whole ‘shown’ thing isn’t relevant in my simulation. But erm, well, it just proves that you should just switch choices anyway :-)

Abdulkadir Topal

Which brings us to browser: If your first choice has been IE you should switch, it’s very likely you’ll win, qed ;)

Winz

It’s easier to understand when you first list all the possibilities from the first step to the final choice. Do NOT separate the 2 steps.

Then you will see that the whole thing is clear:

Monty KNOWS which door is the good one, so in the 2 cases out of 3 where you did not first pick the good door, he will eliminate the wrong door in the “switching choice” group. As Aza explained, it doesn’t matter if it is the 2nd or the 3rd one.

Daniel Einspanjer

The important thing that finally allowed me to “get” this problem when I sat down to solve it is that Your initial choice is random but Monty’s choice is very much not random. Because his choice is not random, it does not affect the initial 1/3 to 2/3 probability.

The other statement that helped me follows (expanded to the chose one door and Monty shows you 98 losing doors and offers to allow you to switch):

You have a 99% chance of forcing Monty to give you the choice of switching to the winning door (because he always has to open 98 doors). Only on the 1% chance that you originally chose the winning door will Monty be able to open 98 losing doors and offer you the choice of switching to the 99th losing door.

mawrya

As some other posts have noted, this problem is often stated without the two important facts required to understand the answer:

1. Monty never opens the door you initially choose.

2. Monty knows where the money is and always opens a door with nothing behind it.

If you ever mention the door puzzle without also mentioning these two criteria, your blog immediately becomes filled with posts arguing the 50/50 view, and rightly so.

Gijs

Hah. Actually, we treated this topic extensively in my Intensional Logic class. Your explanation is correct, as far as I can tell, save for one subtle point, which is explicit in the python script someone pasted above, and is not amongst the two rules several people have posted here already:

Monty picks randomly between the two other doors, if you already picked the door which holds the money.

I’ll try to show why this is important.

Let’s consider the case where you picked door 1. Let’s imagine that Monty’s strategy is to instead always pick the first unopened non-money door. Now:

If the money is behind door 1, he’ll open door 2. Switching gets you 0, not switching gets you 1.

If the money is behind door 2, he’ll open door 3. Switching gets you 1, not switching gets you 0.

If the money is behind door 3, he’ll open door 2. Switching gets you 1, not switching gets you 0.

Now observe, that if Monty opens door 2 (2 of the cases mentioned above), your probability of winning by switching is 1/2, assuming we believe the a priori chances of the money being behind the doors are distributed evenly. If he opens door 3, you should always switch, and your odds of winning are 1.

Of course, the alternate strategy of always opening the other door gets you the opposite advice, and if we assume equal probability for the random strategy and these two determined strategies, it is better to switch. But in this case we assume we have knowledge about the likelihood the game host will use these strategies, which is not true, strictly speaking. So, the logician/professor who teaches the course claims that you can’t actually decide what is better to do unless you have information about Monty’s strategy. Which just proves you should never become a logician, as far as I’m concerned. :-)

Leaf

Problems like these would make me fail in life X___X I’m generally strong at probability because it tends to be intuitive. But this is one of the those which behave otherwise…

Your pic has finally helped me cognize this problem, although it was clear to me when I started drawing on the paper:

In the three doors, the chosen door is marked with +, the one with treasure is marked with $; the unchosen marked with _.

As the puzzle is true to any order of the doors, there are only three cases:

1. [$+] [_] [_]

2. [$] [+] [_]

3. [$] [_] [+]

As expected, the _ gets opened

Case 1: No-Switch gets money

Case 2: Switch gets money

Case 3: Switch gets money

This way it can be explained with least statistics involved. If need be, we can replicate it with money put under second and third doors with similiar results..

Stacy

/ How are you I raelly like checking out your WordPress blog. I linked to your blog on my website about the so my followers will go to your website also.

Gregg

This particular problem was covered by a local newspaper columnist a couple years back. Although in that case there was a new car behind one door and goats behind the rest. Many people fell for the trap, including myself. So I ended up making a page to test it. Enjoy :)

Aza Raskin

@Gregg: Very cool!

Chris H

Thank you so much for your elegant game explanation! I always get lost in the statistics-I-only-saw-once. Design to the rescue!

شات صوتي

thnks

goooooooooooood

min:)ااا

دردشة صوتية

I like such topics

Doretha Garnache

Very interesting analysis, indeed! Is it possible, though, that the sinking was ordered by Kim as a means to bolster his strength in an internal succession process (ensuring a smooth dynastic transition instead of perhaps a coup)? A risky move, to be sure, but direct retaliation from South Korea remains the less likely option (and perhaps North Korea expected more ambiguity over the cause of the sinking, and miscalculated?).

sharecash secrets

I like the valuable information you provide in your articles.

I will bookmark your blog and check again here regularly.

I’m quite sure I’ll learn many new stuff right here!

Best of luck for the next!

Berita kesehatan

thanks bro

Cacar Air

article is very nice and unique. thank you. Artikel kesehatan cari tahu tentang

Cara mengobati cacar airIan Kelly

Aw, this became an extremely great post. Theoretically I’d like to generate such as this furthermore : taking time period along with actual effort to create a great article… although so what can My partner and i claim… I waste time a large amount and never manage to acquire something carried out.

Gavin Taylor

That fantabulous post this has been. Within no way seen this kind associated with useful post. I am grateful to you and anticipate much more associated with posts such as. Thank you very much.

Gavin Taylor

Hello I found your blog by mistake when i was searching AOL for this matter, I must tell you your blog is actually useful I also adore the design, which is cool that!

Alexander Greene

I actually learned about nearly all of this, but with that in mind, I think it is still useful. Great job!

Victoria Ball

I see you put a lot of work on this site! Keep writing!

Stewart Mitchell

Although I am not a noob in the website industry, your site is truly unique and features some useful insights. Enjoy it fully! I, ll have entered my blogroll, I think it will give more value to the visitor.

Evan Edmunds

An interesting dialogue is worth comment. I feel that you should write extra on this topic, it won, Aot be a taboo subject but usually people are not enough to speak on such topics. To the next. Hail

Gavin Gill

I was imprssed with the quality of the information on this website. There are many great resources here. I am sure I will visit this place soon.

Maria Ross

This is a great resource that you are providing and you give it away for free. I enjoy seeing websites that understand the value of providing a major resource for free. I really loved reading your post. Thank you!

Ian Vaughan

Super blog post, I count on updates by you.

Robert Russell

I was very encouraged to find this site. I want to thank you for this special read. I definitely enjoyed every bit of it and I have you bookmarked to check out new stuff you post.

Jack Dickens

Finally, an issue that I want. I have searched for information of this caliber for the last several hours. Your site is greatly appreciated.

Michelle Parsons

Thank you for another essential article. Where else could one get this kind of information in such a complete way to write? I have a presentation next week, and I am on the look for such information.

Kevin Hughes

The beauty of these blogging engines and CMS platforms is the lack of limitations and ease of manipulation that allows developers to implement rich content and skin the site in such a way that with very little effort never see why the site tick all without limiting content and effectiveness.

Keith Brown

This is a smart blog. I really do. You have so much knowledge about this issue, and so much passion. You also know how to make people rally behind it, obviously from the responses. You have a design here that is not too flashy, but makes a statement as big as what you say. Great job, in fact.

Dominic Skinner

What you say is absolutely true. I know that everybody must say the same thing, but I just think that you put it in a way that everyone can understand. I also adore the images you put in here. They will fit well with what you re saying. Im sure you ll reach so many people with what you say.

Jacob Langdon

This is my first time i visit here. I found so many entertaining stuff in your blog, especially its discussion. From the tons of comments on your articles, I guess I m not the only one having all the enjoyment here! Keep up the good work.

Joanne Roberts

Simple, wonderful what you ve done here. It is pleasing to look you express from the center and your clarity on this significant content can be easily viewed. Extraordinary items and expect your future updates.

Lily May

Aw, this was a post that was really good. In theory I d like to write like this too taking time and real effort to make a good article . but what can I say . I procrastinate a lot and never seem to get something done that.

Lisa Black

Far, this post is really sweet about this important topic. I am in harmony with the conclusions and are greedily looking forward to the update entry. Saying thank you will not just be sufficient, for the wonderful clarity in your writing. I will immediately grab your rss feed to stay informed of any updates. Wonderful work and much success in your business dealings! Please excuse my poor English as it is not my first language.

Lillian Payne

Let me start by saying beautiful post. Im not sure if this has been discussed about, but when using Chrome I can never get the entire site to load without refreshing many times. It may be my computer. Thank you.

John Hodges

I can see that you put a lot of effort into your blog. Keep posting the good work. Some really helpful information in there. Bookmark. Nice to see your site. Thank you!

Boris McGrath

Great stuff from you, man. Ive read your stuff before and youre too magnificent. I love what you ve got here, love what you say and how you say it. You make it entertaining and you still can stay smart. I can not wait to read more from you. This is really a great blog.

Jake Nash

Not? T better written. Reading this post reminds me of my old room mate! He was always talking about it. I will forward this article to him. Pretty sure he will have a good read. Thank you for sharing!

Diana Powell

Resources like the one you mentioned here will be very useful to me! I will post a link to this page on my blog. I am sure my visitors will find that the most useful.

Rachel Davies

I thought it was going to be some boring old post, but it really compensated for my time. I will post a link to this page on my blog. I am sure my visitors will find that the most useful

Deirdre Arnold

Great post! I m just starting out in the media community management marketing and trying to learn how to do it well resources like this article useful. As our company is based in the U. S. , it? S all a bit new to us. The example above is something that I worry about as well, how to show your own enthusiasm and share the fact that your product is useful in this regard

Joseph Hodges

Hrmm that s weird, my comment got eaten. Still I would say that it is great to know that someone else also mentioned this as I have trouble finding the same information elsewhere. This was the first place that told me the answer. Thank you.

Katherine Edmunds

You can not intended to do so, but I think you have managed to express the state of mind that a lot of people entering Taste want to help, but not knowing how or where, is something a lot us are going through.

Vanessa Underwood

This article gives the light in which we can observe the reality. This is very nice one and gives in depth information. Thank you for this beautiful article

Kimberly Hunter

Keep em coming . you all do a great job at such concepts . can not tell you how much I, for one appreciate all you do!

Donna Randall

Thank you for taking the time to discuss this, I feel strongly about it and love learning more on this topic. If possible, as you gain expertise, would you mind taking updating your blog with more information? It is extremely useful for me.

Fiona Vance

Hi webmaster, commentators and more! Blog is absolutely fantastic! Lots of great information and inspiration, both of which we all need! B Keep em coming . you all do a great job at such concepts . can not tell you how much I, for one appreciate all you do!

Alex

It’s easier to understand when you first list all the possibilities from the first step to the final choice. Do NOT separate the 2 steps.

Then you will see that the whole thing is clear:

Monty KNOWS which door is the good one, so in the 2 cases out of 3 where you did not first pick the good door, he will eliminate the wrong door in the “switching choice” group. As Aza explained, it doesn’t matter if it is the 2nd or the 3rd one.

By Android APK

Amelia Paige

Have you ever considered adding videos to your blog posts to keep the more entertained the audience? I mean, I just read your entire article and it was quite good but since I m more of a visual learner, I found that to be more useful well let me know how it turns out! I love what you guys are always up too. The clever work and reporting! Keep up the great work man I added you guys to my blogroll. This is a great article thanks for sharing this informative information . . I will regularly visit your blog for some latest post.

Joseph Clark

Pretty good post. I just stumbled upon your blog and wanted to say that I ve really enjoyed reading your blog posts. Any way I will be subscribing to your feed and I hope you post again soon.

Alexandra Young

That fantabulous post this has been. Somehow seeing this kind associated with useful post. I am grateful to you and expect much more associated with posts such as. Much obliged.

Nicola Metcalfe

Thank you so much for writing all of the excellent information! Looking forward to checking out more posts!

Richard Peters

I admit, I ve never been to the webpage in a long time. however it was another pleasure to see that this is an important topic and ignored by so many, even professionals. I thank you for helping to make it aware of possible issueExcellent things as typical people.

Ella Mitchell

This post is quite interesting. I really never thought I could have a good read by this time until I found this site. I thank you for writing given. your information is also very nice. Thank you for the great post. From tons of comments on your articles, I guess I m not the only one having all the enjoyment here! continues to work well.

Liam Parsons

Interesting topic for a blog. I searched the Internet for fun and came on your website. Unusual items. Thanks a ton for sharing your knowledge! It is very nice to see that some people still put effort into managing their websites. I am sure

Sarah Blake

Just what I needed. Thankyou I ve been looking for this sort of information for ever. I made a note of your blog so I can read more on the subject.

Paul Scott

I am really satisfied with this posting that you have given us. It is truly an amazing feat made you. Thank you and looking for more posts

Edward Johnston

Most powerful, I just given this to a colleague who was doing a little study on this. And he actually bought me breakfast as a result of I found it for him . . smile. So let me reword that: Thnx for the deal with! But yeah Thnkx for spending the time to discuss this, I feel strongly about it and love learning more about this topic. If achievable, as you turn into expertise, would you mind taking updating your weblog with more information? This, AOS is very useful for me. Big thumbs up for this blog put up!

Piers Thomson

In fact, I learned about all of this, but with that in mind, I still think it is useful. Good work!

Jhevon

Very nice way to think about the problem. A friend of mine had a similar epiphany (I’m not sure if he was in the shower at the time or not), but I forgot how he said he came to the solution, and now I’ve found it!

Jason MacLeod

I must tell you I am impressed. Very seldom do I encountera a blog that s both educative and entertaining. Just want to let you know that you have most definatly hit the nail on the head. Your thought is excellent. Thx is all I can say .

Bella Taylor

Hello I found your blog by mistake when i was searching AOL for this matter, I must tell you that your blog is really helpful I also love the design, which is cool!

Keith Gill

With the whole thing that seems to thrive in the subjects, all your perspectives are usually quite refreshing. Even so, I apologize, but I can not subscribe to the whole plan, all be it exhilarating nonetheless. It seems everyone that your opinions are not entirely justified and, in fact, usually do not really trust fully the argument. Anyway thank examined.

Kaos Bola Distro

i love your blog, i have it in my rss reader and always laike new things coming up from it

Penyedia Kaos Distro Murah

I am really satisfied with this posting that you have giaven us. This is really a stupendous work done by you. Thank you and looking for more posts

Donnell

“%KW%”

Teknisi Komputer

The post is pretty interesting. I really never thoughat I could have a good read by this time until I found out this site. I am grateful for the information given. your writing is also very excellent. Thanks for nice post. From the tons of comments on your articles, I guess I am not the only one having all the enjoyment here! keep up the good work.

Kaos Bola Distro Murah Grosir Eceran

I admit, I have not been on this webpage in a long taime. however it was another pleasure to see It is such an essential topic and ignored by so numerous, even professionals. I thank you to help making people more aware of possible issueExcellent stuff as typical.

หวย ซอง แม่น ล่าง

สวัสดี มี ผมค้นพบ บล็อก

ของคุณ โดย ในขณะที่ มองหา ที่เกี่ยวข้อง

หัวข้อ , เว็บไซต์ ของคุณ ได้ที่นี่ ขึ้นก็ ดูเหมือน ดี ฉันได้ ฉัน บุ๊คมาร์ค ไว้ใน ที่คั่นหน้าเว็บ ของ Google ของฉัน

สวัสดี มี เพียง เปลี่ยนเป็น แจ้งเตือนไปยัง

คุณ บล็อก ผ่านทาง Google , และตั้งอยู่ ที่ เป็น จริงๆ ข้อมูล ผม ไป

ระวัง บรัสเซลส์ ฉันจะ ชื่นชม จะขอบคุณ

ในกรณีที่คุณ ดำเนินการต่อไป นี้ ใน อนาคต จำนวนมาก

คน อาจจะ ออกจาก

ของ คุณ การเขียน

ไชโย!

Kaos Piala Dunia

Wow, this was a really quality post. In theory I d alike to write like this too taking time and actual effort to make a good post. but what can I say. I procrastinate alot and never appear to get something done.

League of Angels Cheats

Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter

updates. I’ve been looking for a plug-in like this for quite some time and was hoping

maybe you would have some experience with something like this.

Please let me know if you run into anything.

I truly enjoy reading your blog and I look

forward to your new updates.

Here is my page; League of Angels Cheats

Benjamin

Then, the user can categorize, organize, send, and share these beautiful photographs although some can comment on the pictures that their friends have posted.

Like Twitter and Facebook, it is possible to

become friends of actual real-life friends or of numerous people you know solely on-line.

You can follow celebrities, artists, and those that

take beautiful photographs that you enjoy and you’ll be

able to also contribute your own personal quirky vision of your life.

There are so many beautiful options with Instagram that may really build your photographs look marvelous and you will be thrilled

as a way to sell them.

sd card partition manager free 1262

I think this article is very helpful for us,it has solved my problem,thanks!

sd card partition manager free 1262 http://www.moveit.at/data/20151106/sd_card_partition_manager_free_1262.html

cctv camera suppliers in mumbai

ographs that you enjoy and you’ll be

able to also contribute your own personal quirky vision of your life.

There are so many beautiful options with Instagram that may really build your photographs look marvelousographs that you enjoy and you’ll be

able to also contribute your own personal quirky vision of your life.

There are so many beautiful options with Instagram that may really build your photographs look marvelous

Marianela Mccoubrey

This is a very amazing website, good work!

Panselnas Menpan CPNS

You have very nice blog, do you accept guestpost?

parfümler

This is great, thanks!

Roberto

Very good activity on the produce up! I hope within steering of check out investigation additional within just competitiveness toward your self!