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← 05-42 Tuesday Solution

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Showing Revision 1 created 05/13/2012 by Amara Bot.

  1. If I go ahead and execute this and print the result,
  2. it comes out: 13/27.
  3. Wow! Where did that come from?
  4. So that' surprising--first of all, it's not 1/3,
  5. which you might have thought should be the answer
  6. if you believe the argument that Tuesday doesn't matter.
  7. And secondly, not only is it not 1/3,
  8. but it's much closer to 1/2 than it is to 1/3.
  9. So just having the birthday there really changed things a lot.
  10. How did that happen? Well, I wrote up a little function here
  11. to report my findings, and here's its arguments.
  12. You can give it a bunch of cases that you care about--
  13. the predicate that you care about
  14. and whether you want the results to be verbose or not.
  15. And it just prints out some information--
  16. and, by the way, as part of this,
  17. I also looked at the question of
  18. what's the probability of two_boys,
  19. given that there's one boy born in December
  20. so I threw that in as well.
  21. And here's the output I get: 2 boys, given 1 boy is 1/3;
  22. 2 boys, given at least 1 boy born on any day is still 1/3;
  23. and born on Tuesday is 13/27,
  24. and born in December is 23/47.
  25. Now, I can turn on the verbose option to report
  26. In that case, here's what I see:
  27. The probability of 2 boys, given at least 1 boy--
  28. born on Tuesday--is 13/27.
  29. And here's the reason--at least 1 boy, born on Tuesday,
  30. has 27 elements--and there they are--
  31. and of these, 13 are 2 boys--and there they are.
  32. And so, you can't really argue with that.
  33. You can go through and you can make sure that that's correct,
  34. and you can look at the other elements of the sample space
  35. and say no, we didn't miss any--
  36. so that's got to be the right answer.
  37. It's not quite intuitive yet, and
  38. I'd like to define my report function
  39. so that it gives me that intuition
  40. but right now, I don't have the right visualization.
  41. So I've got to do some of the work myself.
  42. And here's what I came up with:
  43. We still have the four possibilities that we showed before
  44. but now we're interested, not just in boys--
  45. we're interested in boys born on Tuesday.
  46. So there's going to be some others over here
  47. where there's, say, boy born on Wednesday,
  48. along with some other partner--
  49. maybe a boy born on Saturday.
  50. But we're not even considering them; we're throwing all those out.
  51. We're just considering the ones that match here.
  52. And like before, we draw 2 circles:
  53. one of the right-hand side of the event--
  54. of the conditional probability.
  55. And so how many of those are there?
  56. Well, there's 7 possibilities here
  57. because the boy has to be born on Tuesday--
  58. there's only 1 way to do that--but there's 7 ways for the girls to be born.
  59. So there's 7 elements of the sample state there;
  60. likewise, 7 elements over here.
  61. Now how many elements over here?
  62. Well here, either one of the 2 can be a boy born on Tuesday.
  63. So really, we should draw this state
  64. as either a boy born on Tuesday, followed by another boy
  65. or a boy, followed by a boy born on Tuesday.
  66. And how many of those are there? Well, there's 7 of these
  67. by the same argument we used in the other case,
  68. and of these, there's also 7
  69. but now I've double-counted because in one of these 14 cases
  70. is a boy born on Tuesday, followed by a boy born on Tuesday.
  71. So I'll just count 6 here.
  72. And so now it should be clear: 7, 14, 21, 6, 27.
  73. There's 27 on the right-hand side,
  74. and then what's the probability of 2 boys,
  75. given this event of at least 1 boy born on Tuesday?
  76. Well, 2 boys--that's here--so it's 13 out of the 27.
  77. So that's the result.
  78. Seems hard to argue with.
  79. Both the drawing it out with a pen
  80. and the computing worked out to the same answer.
  81. Now why is it that we have a strong intuition
  82. that, knowing the boy born on Tuesday
  83. shouldn't make any difference?
  84. I think the answer is because we're associating
  85. that fact with an individual boy.
  86. We're like taking that fact
  87. and nailing it on to him--and it's true.
  88. If we did that, that wouldn't make any difference.
  89. But, in this situation, that's not what we're doing.
  90. We're not saying anything about any individual boy.
  91. If we did that, the computation wouldn't change.
  92. Rather, we're making this assertion
  93. that at least one was born on Tuesday--
  94. not about boys, but about pairs.
  95. And we just don't have very good intuitions
  96. about what it means to say something
  97. about a pair of people,
  98. rather than about an individual person,
  99. and that's what we did here--
  100. and that's why the answer comes out to 13/27.