TED-Ed Frog Riddle
October 10th, 2016 at 10:25:32 PM
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I enjoy reading the articles on Wizard of Odds/Vegas because learning about the math behind the games is fun to me. (And I'm not a math expert!)
Recently, I was obsessed with the TED-Ed frog puzzle:
https://www.youtube.com/watch?v=cpwSGsb-rTs
Long ago, I read about the Monty Hall Paradox and fully understood it. However, I thought this frog puzzle was a different kind of reasoning. So, I present to you my flawed reasoning, then my probably-more-accurate reasoning.
FLAWED REASONING
I initially thought that knowing one of the frogs in the clearing is male didn’t change the probability of the other frog being female. I thought the unknown frog still had a 50-50 chance of being female.
My reasoning was that if you changed the frogs to a deck of cards, you could see that the chance of getting a female frog was closer to 50% than 67%. A red-suited and black-suited card would represent a male and female frog, respectively.
—There are 52 cards: 26 red and 26 black.
—You need to pick a black card to survive.
—The card on the tree stump is unknown.
—In the clearing are two cards—one of which is exposed to be red.
Because you don’t know what's the card on the tree stump, it has a 26 in 51—or 50.98% chance—of being black. (You DO know the exposed card is red, so there’s now only 51 unknown cards.)
—If the card on the tree stump is red, and the known card is red, the chance of the unknown clearing card being black is 26 in 50, or 52%.
—If the card on the tree stump is black, and the known card is red, the chance of the unknown clearing card being black is 25 in 50, or 50%.
—But you don’t know what’s the card on the tree stump, so the unknown clearing card could be either 50% or 52% black.
So do you take the certain 50.98% chance of the tree stump card being black?
Or take the unknown clearing card that has either a 50% or 52% chance of it being black?
PROBABLY-MORE-ACCURATE REASONING
I had an epiphany that my original reasoning is wrong. I thought this puzzle is not like the Monty Hall Paradox because there’s no host manipulating the options. However, I realized that if I were to manually test the probability by dealing two cards to the clearing, I wouldn’t be able to use the results from two black (“female”) cards because one card has to be revealed as a red (“male”). Therefore, I (“the host”) WOULD be manipulating the outcome!
And my manual test bore that result!
I used the Wizard of Odds' blackjack game to deal me two cards. It uses six decks of cards that are shuffled after each hand, so there is equal chance of being dealt a red or black card.
If I received two black (female frog) cards ("Bb" in the results below), I didn't count that in my running total. A red/red ("Rr"=two male frogs) result or either a black/red or black/red ("B") combination result would initiate a new line in my list.
After 100 non-Bb results, I tallied the results. Amazingly (or rather, statistically accurately), 67 were male/female combinations, and 33 were male/male.
The overall percentages were pretty close to 25% for each combination in the sample space: 27% female/female, 49% male/female or female/male, and 24% male/male.
The mental obstacle is that it’s counter-intuitive to think that when you learn one of the two frogs is male, it increases your chance that one is female. However, this is kind of a mental trick. If you’re aware of the concept of “sample space,” knowing that one is male actually reduces the chance of one frog being female from 3 in 4 (75%) to 2 in 3 (67%). Of course, this is still better than the 50-50 chance with the frog on the tree stump. So there’s a bit of trickery going on in the presentation, because without ANY information about the frogs, of course you’d lick two frogs instead of one!
As a side note, I finished ahead 19.5 units in blackjack. I attribute this to favorable house rules, perfect strategy, and good luck.
Not including black/black combinations:
67 black and red combinations 67%
33 red/red 33%
Including black/black combinations:
37 black/black 27%
67 black and red combinations 48.9%
33 red/red 24.1%
137 random combinations
1. B
2. B
3. Rr
4. B
5. B
6. B
7. B
8. Bb. Bb. Bb. Rr
9. B
10. B
11. B
12. B
13. Rr
14. Bb. Rr
15. B
16. Bb. Bb. B
17. B
18. Rr
19. Bb. B
20. Bb. Bb. B
21. B
22. Bb. Rr
23. B
24. Rr
25. Rr
26. Rr
27. Bb. Bb. Bb. Rr
28. Bb. Rr
29. B
30. Bb. B
31. B
32. Bb. B
33. Rr
34. Bb. Bb. Rr
35. Rr
36. B
37. Bb. B
38. B
39. Rr
40. Rr
41. B
42. Bb. Bb. B
43. B
44. B
45. B
46. Rr
47. B
48. Bb. B
49. Bb. Rr
50. Bb. B
51. Bb. Rr
52. B
53. B
54. Bb. Rr
55. B
56. B
57. B
58. Rr
59. B
60. Rr
61. Rr
62. Rr
63. Rr
64. B
65. B
66. B
67. B
68. B
69. B
70. Rr
71. Bb. B
72. Bb. Bb. B
73. Rr
74. B
75. B
76. Rr
77. B
78. Bb. Rr
79. B
80. B
81. B
82. Bb. B
83. Bb. B
84. B
85. Bb. Bb. B
86. B
87. B
88. B
89. Bb. B
90. Bb. B
91. Rr
92. B
93. Bb. Rr
94. B
95. Rr
96. B
97. B
98. Rr
99. B
100. B
Recently, I was obsessed with the TED-Ed frog puzzle:
https://www.youtube.com/watch?v=cpwSGsb-rTs
Long ago, I read about the Monty Hall Paradox and fully understood it. However, I thought this frog puzzle was a different kind of reasoning. So, I present to you my flawed reasoning, then my probably-more-accurate reasoning.
FLAWED REASONING
I initially thought that knowing one of the frogs in the clearing is male didn’t change the probability of the other frog being female. I thought the unknown frog still had a 50-50 chance of being female.
My reasoning was that if you changed the frogs to a deck of cards, you could see that the chance of getting a female frog was closer to 50% than 67%. A red-suited and black-suited card would represent a male and female frog, respectively.
—There are 52 cards: 26 red and 26 black.
—You need to pick a black card to survive.
—The card on the tree stump is unknown.
—In the clearing are two cards—one of which is exposed to be red.
Because you don’t know what's the card on the tree stump, it has a 26 in 51—or 50.98% chance—of being black. (You DO know the exposed card is red, so there’s now only 51 unknown cards.)
—If the card on the tree stump is red, and the known card is red, the chance of the unknown clearing card being black is 26 in 50, or 52%.
—If the card on the tree stump is black, and the known card is red, the chance of the unknown clearing card being black is 25 in 50, or 50%.
—But you don’t know what’s the card on the tree stump, so the unknown clearing card could be either 50% or 52% black.
So do you take the certain 50.98% chance of the tree stump card being black?
Or take the unknown clearing card that has either a 50% or 52% chance of it being black?
PROBABLY-MORE-ACCURATE REASONING
I had an epiphany that my original reasoning is wrong. I thought this puzzle is not like the Monty Hall Paradox because there’s no host manipulating the options. However, I realized that if I were to manually test the probability by dealing two cards to the clearing, I wouldn’t be able to use the results from two black (“female”) cards because one card has to be revealed as a red (“male”). Therefore, I (“the host”) WOULD be manipulating the outcome!
And my manual test bore that result!
I used the Wizard of Odds' blackjack game to deal me two cards. It uses six decks of cards that are shuffled after each hand, so there is equal chance of being dealt a red or black card.
If I received two black (female frog) cards ("Bb" in the results below), I didn't count that in my running total. A red/red ("Rr"=two male frogs) result or either a black/red or black/red ("B") combination result would initiate a new line in my list.
After 100 non-Bb results, I tallied the results. Amazingly (or rather, statistically accurately), 67 were male/female combinations, and 33 were male/male.
The overall percentages were pretty close to 25% for each combination in the sample space: 27% female/female, 49% male/female or female/male, and 24% male/male.
The mental obstacle is that it’s counter-intuitive to think that when you learn one of the two frogs is male, it increases your chance that one is female. However, this is kind of a mental trick. If you’re aware of the concept of “sample space,” knowing that one is male actually reduces the chance of one frog being female from 3 in 4 (75%) to 2 in 3 (67%). Of course, this is still better than the 50-50 chance with the frog on the tree stump. So there’s a bit of trickery going on in the presentation, because without ANY information about the frogs, of course you’d lick two frogs instead of one!
As a side note, I finished ahead 19.5 units in blackjack. I attribute this to favorable house rules, perfect strategy, and good luck.
Not including black/black combinations:
67 black and red combinations 67%
33 red/red 33%
Including black/black combinations:
37 black/black 27%
67 black and red combinations 48.9%
33 red/red 24.1%
137 random combinations
1. B
2. B
3. Rr
4. B
5. B
6. B
7. B
8. Bb. Bb. Bb. Rr
9. B
10. B
11. B
12. B
13. Rr
14. Bb. Rr
15. B
16. Bb. Bb. B
17. B
18. Rr
19. Bb. B
20. Bb. Bb. B
21. B
22. Bb. Rr
23. B
24. Rr
25. Rr
26. Rr
27. Bb. Bb. Bb. Rr
28. Bb. Rr
29. B
30. Bb. B
31. B
32. Bb. B
33. Rr
34. Bb. Bb. Rr
35. Rr
36. B
37. Bb. B
38. B
39. Rr
40. Rr
41. B
42. Bb. Bb. B
43. B
44. B
45. B
46. Rr
47. B
48. Bb. B
49. Bb. Rr
50. Bb. B
51. Bb. Rr
52. B
53. B
54. Bb. Rr
55. B
56. B
57. B
58. Rr
59. B
60. Rr
61. Rr
62. Rr
63. Rr
64. B
65. B
66. B
67. B
68. B
69. B
70. Rr
71. Bb. B
72. Bb. Bb. B
73. Rr
74. B
75. B
76. Rr
77. B
78. Bb. Rr
79. B
80. B
81. B
82. Bb. B
83. Bb. B
84. B
85. Bb. Bb. B
86. B
87. B
88. B
89. Bb. B
90. Bb. B
91. Rr
92. B
93. Bb. Rr
94. B
95. Rr
96. B
97. B
98. Rr
99. B
100. B
October 11th, 2016 at 4:43:05 AM
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Thank you for the Ted Ed link.
I know that male frogs croak but did not know it was unique to the male.
Conditional probability ... h'mmm what if we know the average roll is 8.4 and ... oh yeah, we've been through that one. I think it was last demonstrated after a newbie lesson and they served champagne to the woman who broke the record. Not a good time to be focusing on 8.4 and thinking of going dark side when records are being broken.
I know that male frogs croak but did not know it was unique to the male.
Conditional probability ... h'mmm what if we know the average roll is 8.4 and ... oh yeah, we've been through that one. I think it was last demonstrated after a newbie lesson and they served champagne to the woman who broke the record. Not a good time to be focusing on 8.4 and thinking of going dark side when records are being broken.
