Probability Problem

• Kueh wrote on 2011-09-07 09:45
  (Inspired by Kitae's last thread like this.)
  
  This question comes in two parts, and it took me a bit till I accepted the answers I was told. At first, I was really confused, and was making this thread just to ask if someone could explain it to me, but I understood it as I was typing, so I might as well post it.
  
  

  Suppose someone tells you this, "That woman has two children, and at least one of them is a boy."
  (Ignoring intersex/hermaphrodites) What is the probability that she has two sons?

  
  
  And the second question is solved the same way, but the answer comes out to be something much different.
  
  

  Suppose I tell you that I have two children, and one is a boy who was born on Friday. What is the probability that I have two sons?

  
  
  These questions are actually pretty easy, but their answers are different from what seems natural, so it's hard to accept them.
• Conor wrote on 2011-09-07 09:48
  How are they both not 50%?
• Kueh wrote on 2011-09-07 09:50

  Quote from Conor;580962:

  How are they both not 50%?

  
  
  Because I said, "At least one is a boy."
  
  That means there are no situations where I have two girls.
• Conor wrote on 2011-09-07 09:52

  Quote from Lyre;580963:

  Because I said, "At least one is a boy."
  
  That means there are no situations where I have two girls.

  
  Yeah, so the only other situation is a boy and a girl, vs boy and boy.
  so how is not 50%...
  I'm not good at probability, th0. im jus askin
• Kueh wrote on 2011-09-07 09:57

  Quote from Conor;580965:

  Yeah, so the only other situation is a boy and a girl, vs boy and boy.
  so how is not 50%...
  I'm not good at probability, th0. im jus askin

  
  
  Because boy and a girl is twice as likely than boy and boy. So you can either think of it as:
  
  

  
  
  • Boy, Girl
    
  • Girl, Boy
    
  • Boy, Boy
    

  
  
  Which makes the answer kind of obvious, or you can just keep in mind that it's twice as likely to get a boy and a girl than it is to get two boys.
• Wasbewwi wrote on 2011-09-07 10:35
  Uhhh it's 50, 50?
  
  Okay, question 1. At least one is boy, and the other can be boy/girl. Thus, 50% ..
  
  Okay, question 2. Born on Friday (doesn't mean anything, it's there to throw you off of the topic) .. and still, a boy/girl. Thus, 50%.
  
  ???? Did I win?
• Kueh wrote on 2011-09-07 10:55

  Quote from Wasbewwi;580984:

  Uhhh it's 50, 50?
  
  Okay, question 1. At least one is boy, and the other can be boy/girl. Thus, 50% ..
  
  Okay, question 2. Born on Friday (doesn't mean anything, it's there to throw you off of the topic) .. and still, a boy/girl. Thus, 50%.
  
  ???? Did I win?

  
  
  Sorry, no. For the first problem, think of it like this.
  
  Let's say the children are labeled "A" and "B".
  
  Child A can be either a boy or a girl.
  Child B can be either a boy or a girl.
  
  Then your possible answers are:
  
  A[SIZE="1"]girl[/SIZE], B[SIZE="1"]boy[/SIZE]
  A[SIZE="1"]boy[/SIZE], B[SIZE="1"]girl[/SIZE]
  A[SIZE="1"]boy[/SIZE], B[SIZE="1"]boy[/SIZE]
  A[SIZE="1"]girl[/SIZE], B[SIZE="1"]girl[/SIZE]
  
  But I said that at least one is a boy. So that eliminates one possible situation, and the new list of possibilities is:
  
  A[SIZE="1"]girl[/SIZE], B[SIZE="1"]boy[/SIZE]
  A[SIZE="1"]boy[/SIZE], B[SIZE="1"]girl[/SIZE]
  A[SIZE="1"]boy[/SIZE], B[SIZE="1"]boy[/SIZE]
  [S]A[SIZE="1"]girl[/SIZE], B[SIZE="1"]girl[/SIZE][/S]
  
  So, the probability of a given occurrence is the ratio of that occurrence as a ratio of the total number of possible occurrences. We're just looking for boy and boy, so the probability is 1/3.
  
  As for the second problem, the day actually does matter. It's a trick, but not one to throw you off track.
• Domirade wrote on 2011-09-07 12:57
  1 in 3, 1 in 2.
• arklian wrote on 2011-09-07 13:13
  13/27
  
  

  Quote from Domirade;581047:

  1 in 3, 1 in 2.

  
  
  One of the options overlaps.
• Chiyuri wrote on 2011-09-07 20:50
  I have to disagree with that, both situation are 33.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333...%
  
  The solution you took for question one can also apply to question two.. Even if one son is born on a friday.. That son may either be Aboy or Bboy meaning if we errace the possibility of Agirl and Bgirl togeter.. we still have 3 possible situation and only one of them is Aboy and Bboy.
• arklian wrote on 2011-09-07 23:57
  If you take into account the days, the two childern can be born on...
  (Older, Younger)
  Monday-Monday
  Monday-Tuesday
  ...
  ...
  ...
  Sunday, Sunday
  
  For each pair of days, there are 4 combinations of genders...
  (Older, Younger)
  Male, Male
  Male,Female
  Female,Male
  Female,Female
  
  Female,Female cannot be the case as there is at least one boy.
  
  Either the older or the younger can be the boy that is born on Friday.
  If the older is born on Friday, then the possible days it can be are ...
  
  Friday,Monday
  Friday,Tuesday
  ...
  Friday,Sunday
  A total of 7 pairs of days.
  
  If the older is a male, then the two gender pairs can be Male,Male or Male,Female
  
  This leads to 14 different possibilities with 7 of them being two males.
  
  If you repeat it with the boy being born on Friday as the younger child, then it will also be 14 possibilities with 7 of being two males.
  
  Monday,Friday
  Tuesday,Friday
  ...
  Sunday,Friday
  
  Female,Male
  Male,Male
  
  On Friday,Friday however, the Male,Male combinations overlap for the two cases.
  
  So there are 27 different times where one boy is born on a Friday, and 13 times when it is male,male.
• Chiyuri wrote on 2011-09-08 00:09

  Quote from arklian;581371:

  If you take into account the days, the two childern can be born on...
  (Older, Younger)
  Monday-Monday
  Monday-Tuesday
  ...
  ...
  ...
  Sunday, Sunday
  
  For each pair of days, there are 4 combinations of genders...
  (Older, Younger)
  Male, Male
  Male,Female
  Female,Male
  Female,Female
  
  Female,Female cannot be the case as there is at least one boy.
  
  Either the older or the younger can be the boy that is born on Friday.
  If the older is born on Friday, then the possible days it can be are ...
  
  Friday,Monday
  Friday,Tuesday
  ...
  Friday,Sunday
  A total of 7 pairs of days.
  
  If the older is a male, then the two gender pairs can be Male,Male or Male,Female
  
  This leads to 14 different possibilities with 7 of them being two males.
  
  If you repeat it with the boy being born on Friday as the younger child, then it will also be 14 possibilities with 7 of being two males.
  
  Monday,Friday
  Tuesday,Friday
  ...
  Sunday,Friday
  
  Female,Male
  Male,Male
  
  On Friday,Friday however, the Male,Male combinations overlap for the two cases.
  
  So there are 27 different times where one boy is born on a Friday, and 13 times when it is male,male.

  
  
  The day is rather irrevelant to it all...
  a Day doesn't affect if a person is born male or female.
• arklian wrote on 2011-09-08 01:49
  You cannot have male female and female male on the same pair of days, so the answer is close to 1/2.
• Joker wrote on 2011-09-08 02:16
  this kind of problems reminds me of an oldschool probability test its pretty awesome but difficult to understand
  
  It's called the Monty Hall Problem:
  
  

  Suppose you're on a game show, and you're given the choice of three doors:
  
  > Behind one door is a car;
  > behind the others, goats.
  
  You pick a door, say No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
  

  
  Well?
  **I'll put the video of this problem In the spoiler watch it after you make a guess/post**
  
  [SPOILER="Spoiler"][video=youtube;cXqDIFUB7YU]http://www.youtube.com/watch?v=cXqDIFUB7YU[/video][/SPOILER]
• Kueh wrote on 2011-09-08 03:00

  Quote from Chiyuri;581383:

  The day is rather irrevelant to it all...
  a Day doesn't affect if a person is born male or female.

  
  
  No, you're right it doesn't. But it's not the chance that a child I have is born a boy or a girl. That would be 50% no matter what day. The question is, I have two children, and one of them is a boy who was born on friday. That means if you were to list the possible combinations, there could be no situation where friday had no boys born on that day.
  
  Arklian's answer is spot on, and the fraction comes out to being about .48