A classic puzzle showing most people have unreliable intuition about Cond Prob.
Question A: Suppose there’s a club for mothers of exactly 2 kids — no more no less. You meet Alice and you know she has at least one boy. What’s Prob(both boys)?
Question K: You meet Kate (at clubhouse) along with her son. What’s P(she has 2 boys)?
Question K2: You also see the other kid in the stroller but not sure Boy or Girl. What’s P(BB)? This is essentially the same question on P166 [[Cows in the maze]]
Solution A: 4 equi-events BB/BG/GB/GG of 25% each. GG is ruled out, so she is equally likely to be BB/BG/GB. Answer=33%
Solution K: 8 equi-events BB1/BB2/BG1/GB2/BG2/GB1/GG1/GG2. The latter 4 cases are ruled out, so what you saw was equally likely to be BB1/BB2/BG1/GB2. Answer=50%
Question C: Each mother wears a wrist lace if she has a boy and 2 if 2 boys (Left for 1st born, Right for 2nd born). Each mother comes with a transparent (hardly visible) hairband if she has either 1 or 2 boys. There are definitely more wrist laces than hairbands in the clubhouse. If you notice a mother with a hairband, you know she has either 1 or 2 wrist laces. If you see a wrist lace, you know this mother must have a hairband.
C-A: What’s P(BB) if you see a mother with a hairband?
C-K: What’s P(BB) if you see a mother with a wrist lace on the left hand?
Solution C-A: Out of 2000 mothers, 1500 have hairband. 500 have 2 boys. P(BB) = 33%
Solution C-K: 500 have 2 wrist laces; 500 have only a left wrist lace; 500 have only a right wrist lace. P(BB) = 50%
Seeing a wrist lace is not the same as seeing a hairband. The 2 statements are NOT equivalent. Wrist laces (2000) outnumber hairbands (1500). A wrist lace sighting guarantees a hairband, so a wrist lace is more Rare, and a hairband sighting is more Common. Within the clubhouse, 3 out of 4 hairband “tests” are positive, but only 2 out of 4 wrist lace tests are positive.
Applied to original questions…
* Alice wears hairband but perhaps One of her wrists might be naked. If she brings one child each time to clubhouse, we may not always see the a boy.
* Kate wears at least one wrist lace (so we know she has a hairband too).
$ if we randomly “test” Alice for wrist lace on a random hand, she may fail
$ if we randomly “test” Alice for hairband, sure pass.
–> the 2 tests are NOT equivalent.
$$ if we randomly “test” Kate for wrist lace on a random hand, she may fail
$$ if we randomly “test” Kate for hairband, sure pass.
–> the 2 tests are NOT equivalent for Kate either
The wrist-lace-test pass implies hairband-test pass, but the same knowledge object contains additional knowledge. The 2 tests aren’t equivalent.
—– How is Scenario K2 different from A?
–How many mothers are like K2? We need to divide the club into 8 equal groups
* perhaps Kate is from the BB group and you saw the first kid or the 2nd kid
* perhaps Kate is from the BG group and you saw the first kid – BG1
* perhaps Kate is from the GB group (500 mothers) and you saw the 2nd kid – GB2. Now if you randomly pick one hand from each GB mother then 250 of them would show left hand (GB1) and 250 of them would show right hand (GB2). Dividing them into 2 groups, we know Kate could be from the GB2 group.
=} Kate could be from bb1, bb2, bg1, gb2 groups. In other words, all these 4 groups are “like Kate”. They (1000 mothers) all wear wrist lace, but not all having wrist lace are like-Kate — The bg2 (250) and gb1 (250) mothers are not like-Kate
–How many mothers are like Alice? 75% consisting of BB BG GB
^ Spotting a hairband, the wearer (Alice) is equally likely from the 3 groups — BB(33%) BG(33%) GB(33%)
^ Spotting a wrist lace, the wearer (Kate) is more likely from the BB group (50%) than BG(25%) or GB(25%) group.
If I hope to meet a BB mother, then spotting a wrist lace is more valuable “signal” than a hairband. Reason? Out of the 2000 mothers, there are 2000 wrist laces, half of them from-BB. There are 1500 hairbands, and a third of them are from-BB.
Further suppose each twin-BB mother gets 100 free wrist laces (because wrist lace manufacturer is advertising?), and all the BB mothers claim to have a twin-BB. As a result, wrist laces explode. Virtually every wrist lace you see is from-BB.
There are many simple ways of reasoning behind the 33% and 50%, but they don’t address the apparent similarity and the subtle difference between A and K. When would a reasoning become inapplicable? It’s good to get to the bottom of the A-vs-K difference, the subtle but fundamental. A practitioner needs to spot the difference (like an eagle).