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#1
December 22nd, 2010, 11:55 PM
 zengrifter Executive Member Join Date: Nov 2005 Location: SoCal Posts: 10,532
When intuition and math probably look wrong

This brings to mind the sort of advanced thinking that tourney pros may contemplate
when sizing their next bet. It also reminds me of the Monte Hall proposition in 21. zg
...........
When intuition and math probably look wrong
sciencenews.org

I have two children, one of whom is a son born on a Tuesday.
What is the probability that I have two boys?

Gary Foshee, a puzzle designer from Issaquah, Wash., posed this puzzle during his talk this past March at Gathering 4 Gardner, a convention of mathematicians, magicians and puzzle enthusiasts held biannually in Atlanta. The convention is inspired by Martin Gardner, the recreational mathematician, expositor and philosopher who died May 22 at age 95. Foshee’s riddle is a beautiful example of the kind of simple, surprising and sometimes controversial bits of mathematics that Gardner prized and shared with others.

“The first thing you think is ‘What has Tuesday got to do with it?’” said Foshee after posing his problem during his talk. “Well, it has everything to do with it.”

Even in that mathematician-filled audience, people laughed and shook their heads in astonishment.

When mathematician Keith Devlin of Stanford University later heard about the puzzle, he too initially thought the information about Tuesday should be irrelevant. But hearing that its provenance was the Gathering 4 Gardner conference, he studied it more carefully. He started first by recalling a simpler version of the question called the Two Children Problem, which Gardner himself posed in a Scientific American column in 1959. It leaves out the information about Tuesday entirely: Suppose that Mr. Smith has two children, at least one of whom is a son. What is the probability both children are boys?

MORE- http://www.sciencenews.org/view/gene...bly_look_wrong
#2
December 23rd, 2010, 03:29 PM
 QFIT Executive Member Join Date: Jul 2005 Location: NO LONGER HERE Posts: 2,884

Intuition and math don't look wrong.
#3
December 23rd, 2010, 04:06 PM
 assume_R Executive Member Join Date: Dec 2009 Posts: 856

ZG,

If you read through the comments, you'll realize that the intuition that there's a 50% probability of the other child being the boy is actually correct. The article was mistaken. Here are the true possibilities of the second child being a boy, given that one of the children is a boy (credit from the comments):

"
First, let's assume birth order does NOT matter, here are the possibilities for the gender of a child:
1) Boy
2) Girl
so 50% chance of Boy.

If we assume birth order matters, then these are the possible combinations:
1) Boy(known), Boy
2) Boy(known), Girl
3) Boy, Boy(known)
4) Girl, Boy(known)
Still 50% chance of Boy.
"

The article tried to present the answer of 1/3 (33%), which wasn't correct.
#4
December 23rd, 2010, 04:12 PM
 Sucker Executive Member Join Date: Feb 2009 Location: U.S.A. Posts: 1,504

Quote:
 Originally Posted by assume_R The article tried to present the answer of 1/3 (33%), which wasn't correct.
You need to go back and carefully STUDY the article. It presents the answer of 13/27, which is ABSOLUTELY correct.
#5
December 23rd, 2010, 04:34 PM
 assume_R Executive Member Join Date: Dec 2009 Posts: 856

Sucker,

I was referring to the simpler problem given in the article. I studied the article a bit more, and I still stand by what I said. I will explain myself fully, and please feel free to point out the problem in my logic.

*********************

I will use 3 names in my reasoning: John, Matt, Sally.

The problem is that the article doesn't take into account that we are "given" that John was born on Tuesday. That is the assumption upfront. We already have John born on Tuesday. We also know that John has a sibling. The article also doesn't take into account that the order in which they are born is not important. So here are my possibilities, given that John is born on a Tuesday:

1. John: Tuesday, Matt: Sunday
...
7. John: Tuesday, Matt: Saturday

8. John: Tuesday, Sally: Sunday
...
14. John: Tuesday, Sally: Saturday.

That is 14 possibilities, of which 7 of them have a male sibling (Matt), and 7 of them have a female sibling (Sally). Hence 50%.

*******************

Now, regarding the article's logic.

If you look at their chart http://www.sciencenews.org/view/down...WORKING_IT_OUT you can see that they would consider John and Sally both born on Tuesday as separate events that should be counted twice towards the total. Because their logic is that if John is younger, that's 1 event, and if Sally is younger, that's another event. In that case, yes, their reasoning is correct. However, nowhere in the original problem implies that John and Sally both being born on Tuesday should be separate events, because "what if John is younger, or what if Sally is younger" should imply counting the situation of both John and Sally being born on a Tuesday twice. That is where they are incorrect.
#6
December 23rd, 2010, 06:57 PM
 Automatic Monkey Banned Join Date: May 2006 Location: Connecticut Posts: 5,171

It's a problem of English. What's confusing is that it is unclear if the "one" used in the sentence represents the number 1, or what Germans would say as "man" - "One eats breakfast in the morning and dinner at night," und so wieder.

In the second sense of the word, there is an increased probability that the second child is also a son born on Tuesday, because of the nonzero and non-excluded chance his children are identical twins. Barring that, and using this non-exclusive sense of the word "one" the other child is equally likely to be male or female or born on any day of the week.
#7
December 23rd, 2010, 07:13 PM
 QFIT Executive Member Join Date: Jul 2005 Location: NO LONGER HERE Posts: 2,884

Ungenau. I thought the same and gave no more thought to the question. There simply is no reason to attempt to devine the meaning of an imprecise question. Before you answer a question, you must understand it. It sometimes pisses off people that e-mail me questions when my response is more questions. But, precision is important. Otherwise, there can be multiple answers to a question, all conflicting and yet all arguably correct. Just not correct for the person asking the question.
#8
December 24th, 2010, 01:53 PM
 The Chaperone Senior Member Join Date: Apr 2007 Posts: 350

My intuition says some of this German looks wrong.
#9
December 27th, 2010, 10:39 PM
 Sucker Executive Member Join Date: Feb 2009 Location: U.S.A. Posts: 1,504

Quote:
 Originally Posted by assume_R ZG, If you read through the comments, you'll realize that the intuition that there's a 50% probability of the other child being the boy is actually correct. The article was mistaken. Here are the true possibilities of the second child being a boy, given that one of the children is a boy (credit from the comments): " First, let's assume birth order does NOT matter, here are the possibilities for the gender of a child: 1) Boy 2) Girl so 50% chance of Boy. If we assume birth order matters, then these are the possible combinations: 1) Boy(known), Boy 2) Boy(known), Girl 3) Boy, Boy(known) 4) Girl, Boy(known) Still 50% chance of Boy. " The article tried to present the answer of 1/3 (33%), which wasn't correct.
You're not setting this up correctly -
For THIS problem, there are exactly THREE possibilities, all of which will occur with equal frequency:
1) The first child is a boy, second child is a girl.
2) The first child is a girl, the second child is a boy.
3) Both children are boys.

It's quite clear from this that the chance of two boys can ONLY be 1/3.

Quote:
 Originally Posted by assume_R Sucker, I was referring to the simpler problem given in the article. I studied the article a bit more, and I still stand by what I said. I will explain myself fully, and please feel free to point out the problem in my logic. ********************* I will use 3 names in my reasoning: John, Matt, Sally. The problem is that the article doesn't take into account that we are "given" that John was born on Tuesday. That is the assumption upfront. We already have John born on Tuesday. We also know that John has a sibling. The article also doesn't take into account that the order in which they are born is not important. So here are my possibilities, given that John is born on a Tuesday: 1. John: Tuesday, Matt: Sunday ... 7. John: Tuesday, Matt: Saturday 8. John: Tuesday, Sally: Sunday ... 14. John: Tuesday, Sally: Saturday. That is 14 possibilities, of which 7 of them have a male sibling (Matt), and 7 of them have a female sibling (Sally). Hence 50%. ******************* Now, regarding the article's logic. If you look at their chart http://www.sciencenews.org/view/down...WORKING_IT_OUT you can see that they would consider John and Sally both born on Tuesday as separate events that should be counted twice towards the total. Because their logic is that if John is younger, that's 1 event, and if Sally is younger, that's another event. In that case, yes, their reasoning is correct. However, nowhere in the original problem implies that John and Sally both being born on Tuesday should be separate events, because "what if John is younger, or what if Sally is younger" should imply counting the situation of both John and Sally being born on a Tuesday twice. That is where they are incorrect.
The article does NOT state that John was born on a Tuesday. It says ONE boy was born on Tuesday; so you have to also include the possibility that it was Matt who was born on Tuesday.

There are 27 equal possibilities, of which only 13 consists of two boys:

John: Tuesday
Matt: Sunday

John: Tuesday
Matt: Monday

John: Tuesday
Matt: Tuesday

John: Tuesday
Matt: Wednesday

John: Tuesday
Matt: Thursday

John: Tuesday
Matt: Friday

John: Tuesday
Matt: Saturday

********

Matt: Tuesday
John: Sunday

Matt: Tuesday
John: Monday

Matt: Tuesday
John: Wednesday

Matt: Tuesday
John: Thursday

Matt: Tuesday
John: Friday

Matt: Tuesday
John: Saturday

********

Sally: Sunday
John: Tuesday

Sally: Monday
John: Tuesday

Sally: Tuesday
John: Tuesday

Sally: Wednesday
John: Tuesday

Sally: Thursday
John: Tuesday

Sally: Friday
John: Tuesday

Sally: Saturday
John: Tuesday

********

Sally: Sunday
Matt: Tuesday

Sally: Monday
Matt: Tuesday

Sally: Tuesday
Matt: Tuesday

Sally: Wednesday
Matt: Tuesday

Sally: Thursday
Matt: Tuesday

Sally: Friday
Matt: Tuesday

Sally: Saturday
Matt: Tuesday

Note that the reason that there are only 27 possibilities rather than 28, is the fact that if both boys were born on Tuesday, you can only count that ONCE, rather than twice.

So this article did INDEED give the correct answer to BOTH problems;
1/3 for the Two Child Problem, and 13/27 for the second problem. As it turns out, all those mathematical geniuses are right after ALL!

If you're still having problems understanding this problem, I would suggest that you "Google" and study a similar problem; the "Monty Hall paradox", which involves the same principles, only much simpler to understand.
#10
January 28th, 2011, 10:24 PM
 jopke Member Join Date: May 2010 Posts: 33

Quote:
 Originally Posted by Sucker You're not setting this up correctly - For THIS problem, there are exactly THREE possibilities, all of which will occur with equal frequency: 1) The first child is a boy, second child is a girl. 2) The first child is a girl, the second child is a boy. 3) Both children are boys. It's quite clear from this that the chance of two boys can ONLY be 1/3.
Why does order matter when it is boy/girl and not matter when it is boy boy? It seems to me there are FOUR possibilities:

1) The first child is a boy, second child is a girl.
2) The first child is a girl, the second child is a boy.
3) The first child is the known boy, second child is a boy.
4) The first child a boy, second child is the known boy.

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