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# bonanova

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1. ## Casino Game (big)

I think we are agreeing loudly on this point. The positions of the colors are random [unknown to us] and the combinations of so many reds and so many greens -- it doesn't matter that we used different numbers -- permit an enumeration of all possible, equally likely distributions. What's not 50-50 is the chance that any one particular cup is red or green. Count the number of times in my 26 equally likely color distributions that the first cup [for example] is red. It's 15. Not 13. -- or is green. It's 11. Not 13. Next, you see that in 7 of the 15 red-removed cases any 2 of the remaining 4 cups have the same color. In 5 of the 11 green-removed cases any 2 of the remaining 4 cups have the same color. If all you are saying is that 7/15 is different from 5/11, then the discussion is over and we can get some sleep. [Perhaps erroneously] I took your Case 2 question to mean in all cases when a cup is picked and removed from the group [not a particular case, like a red cup is picked first] does your chance of picking two matching colors differ from your chance of picking two matching colors when all 5 cups are there. My answer to that question is that the chances do not change. This follows from the observation you can make by inspecting the 26 equally likely distributions of 2 colors among 5 cups. Namely, that whatever two cups you inspect, 12/26 of the time they will be the same color. That statement is true whether any or all of the other three cups were removed initially or not. That is, there are no circumstances that force you to pick 2 particular cups that do not have a 12/26 chance of having the same color, simply because every pair of cups in the 26 equally likely cases has a 12/26 chance of a color match. Finally, it appears that if you eliminate the 5-red case [requiring at least 1 green] you will decrease the total distributions from 26 to 25, and the colors-will-match distributions from 12 to 11. That would change the answer for both Case 1 and Case 2 to 11/25 = 0.44.
2. ## I think it's a paradox

This is one for the over 65 crowd. I turn up my Miracle Ear hearing aid and say, "What?"
3. ## kids puzzle

I can think of at least one other common oochie
4. ## The shrewd alchemist

Alex didn't realize that; so to be fair, he should get his 18% and you should get the other 6% -- about \$670 a day ... that sound alright?
5. ## Casino Game (big)

In Case 2, you pick a cup -- and without knowledge this is a random pick -- and then eliminate that cup. Since there are more red than green cups [in the sample space I used, where 5 reds were possible], the odds of picking a red cup first are different from the odds of picking a green cup first, and the odds of getting a pair after picking a red cup differ from the odds of getting a pair after picking a green Not that that really matters, I just made a note of it. In the space of all possible outcomes for Case 2, [that includes 15 times a red cup is picked first and 11 times that a green is picked first] the probability of getting a pair after one cup is eliminated is the same as for Case 1. In either case, it's simply the odds that any two cups are the same color: 12/26 = .461 You could take away three cups at first then ask the odds that the remaining cups are a match. Same result.
6. ## Pair of Aces

What Matt proposed was not that Alex had to get aces on the first two cuts, but only that two successive cuts be aces as he continued thru the deck. The first 30 cards could be non-aces, for example, followed by aces on the 32nd and 33rd cuts. I think that's what you calculated. And now I must confess I haven't done the calculation, which I will take care of, and then post my result. I think the calculation is made easier by considering the probability that a particular ace [hearts, say] is followed immediately by one of the other aces in a well-shuffled deck, then multiply that by four. [not followed or preceded; I think that leads to double counting.]
7. ## kids puzzle

I agree with one thousand. We don't say, for example, twenty and three for 23, nor one hundred and forty and seven for 147. Sorry, I should say, maybe some do say that; I don't.
8. ## Casino Game (big)

Hmmm...? Interesting. No more than 3 green means at least TWO red. Anyway, that aside, this gives 26 distributions of chips: 1 - 0 green, 5 red 5 - 1 green, 4 red 10 - 2 green, 3 red 10 - 3 green, 2 red - but stop here: no more than 3 green are present. Specifically, the distributions are Case 1: "In a row" suggests that perhaps all five are picked in sequence and you hope that two consecutive picks are the same color. [With only two picks, they are always in a row.] But the rest of the wording suggests you have only two picks. I will assume you get only two picks. Case 2: Here you pick one cup and are told what color it is. Then you must pick two other cups. Again since all outcomes are equally likely, we can assume we pick and are told the color of the first cup, and then see in how many of the 26 outcomes cups 2 and 3 [or any other pair of cups] are the same color. We already know that two specific cups have a 12/26 chance of being the same color. Probability again is 12/26 = 0.461
9. ## Dear dead dog gone.

Sorry, my first answer was that lacking fingers he could not open his canteen. So I'm wondering how he could die of a full bladder. Good one. Add his "male"ness to the puzzle and you have a winner.
10. ## How Many Were Going To Saint Ives?

It was also in Die Hard with a Vengeance.
11. ## The spectacular coin toss

11/128 Here's why:
12. ## Weird words III

The words are special because no other English word rhymes with them.
13. ## Pair of Aces

"I bet you can't cut two aces out of a shuffled deck," said Davey to Alex down at Morty's last night. Davey was still sore that Alex had flipped 10 Tails on Monday. "I'm sure I wouldn't take that bet," Alex replied, "I happen to know that my chances are [4/52]*[3/51], and I'll tell ya right now that my dear momma didn't raise no fools!" Matt the mathematician overheard the two boys talkin' and pushed Alex a little farther: "But what if you could continue cuttin' the deck until the cards were all gone? Do you know what yer odds would be to cut consecutive aces, then?" And he sipped his brew while Alex thought about it. After a moment, Alex replied, "Give me a dollar against my penny and yer on!" Was Alex correct about his chances on the first bet? What odds should he have demand to make the second bet fair? Edited for clarity.
14. ## The spectacular coin toss

Great analysis! But ... Take it easy on Matt ... it took me 5 tries to develop his character.
15. ## Troll Problem

We assume the troll knows which road is safe. Ask: "Which road would you have told me yesterday was safe?" He'll either lie about telling you the truth or tell the truth about lying to you. Either way, he'll point to the dangerous road. Then take the other road.
16. ## Wealth of interesting images

Interesting site ... maybe it's well known already? http://lookmind.com/illusions.php
17. ## The spectacular coin toss

There was a bet going on at Morty's last night, about who could toss the most consecutive Tails in 10 flips of a fair coin. Davey went first and came up with H T T H T H T T T H -- 3 Tails in a row Tom was next, and he flipped T T H T H H T T H H -- only 2 Tails in a row, but twice. Next was Pete, who struck out completely with T H T H T H T H T H -- none. Slim was next, starting out with 5 Tails; T T T T T H H T H H -- 5. Then the unthinkable happened: Alex flipped 10 consecutive Tails. Phil grabbed a pencil and tried to figure the odds, but he'd had too many beers. So had Davey, but it didn't stop him from opining that none of them, in any of their lifetimes, would see again what Alex had done. They all agreed, and toasted Alex the rest of the night. Slim felt slighted, saying his sequence was improbable enough for him to at least have gotten a free beer. And Davey and Tom mumbled that it would be tough to repeat what they'd done, too. Finally, Pete claimed that even his result deserved a frosty one, "on the house." But Matt the mathematician -- when he was sober at least -- would have none of the whining. "Anybody could do what you blokes done t'night," he snarled, "but it would take a bazillion years to do what Alex did." Who was right? Departing from my normal MO, here's my answer.
18. ## The shrewd alchemist

Nothing so tricky as all that - no paint, etc. The alchemist's goal was always to turn each atom of lead directly into an atom of gold. There may be a small difference in the weights of those atoms, but that's not the way Alex made his fortune. Here's how it worked.
19. ## Truth in packaging II

The cobbler? When I wuz a kid, I once had my shoes re-souled. And one time I ate some soul-food, but let's not implicate any restaurant owners here.
21. ## The shrewd alchemist

Nope. We need a means by which Alex becomes rich. Answer tomorrow; today's hint: The King had his Royal Department of Weights and Measures weigh the lead Alex received each day, and weigh the gold Alex returned each morning. If they both weighed exactly 1 pound, how could Alex have kept any of the gold?
22. ## A golden oldie

Yup. Misdirection fails again
23. ## Reservoir

Lucid's analysis is the clearest. What you get when you turn on more taps is more flow. Find the flow rates and add them. The taps flow, respectively at rates of 1/2, 1/3, 1/4 and 4 reservoirs per day. Flowing together, [adding their flow rates] the rate is 61/12 reservoirs per day. They will fill 1 reservoir in exactly 12/61 days. That comes to 4 hours, 43 minutes and .. about 16.7213114 seconds.
24. ## The shrewd alchemist

Hint: Helen and an infamous horse.
25. ## more weird words ...

Yup, those all work. Here are the ones I had in mind: [2] Stretched [3] Queueing [4] Witchcraft - that would be in Salem, MA [5] Latchstring - you got it [7] Indivisibility And ... for those who care .... abstemiously: adverb: in a sparing manner; without overindulgence
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