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bonanova

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Posts posted by bonanova

  1. 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.

    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. :blush:

    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.]

  2. CASINO GAME

    There are five cups.

    Under each cup is a paint chip, those little colored cards in hardware stores.

    There are only RED and GREEN paint chips.

    There is AT LEAST ONE red paint chip and NO MORE THAN THREE green paint chips.

    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

    To make it easier to distinguish the colors, use 0 for red, and use 1 for green.

    0 0 0 0 0 - 1 case of 0 green, 5 red

    1 0 0 0 0 - 5 cases of 1 green, 4 red

    0 1 0 0 0

    0 0 1 0 0

    0 0 0 1 0

    0 0 0 0 1

    1 1 0 0 0 - 10 cases of 2 green, 3 red

    1 0 1 0 0

    1 0 0 1 0

    1 0 0 0 1

    0 1 1 0 0

    0 1 0 1 0

    0 1 0 0 1

    0 0 1 1 0

    0 0 1 0 1

    0 0 0 1 1

    1 1 1 0 0 - 10 cases of 3 green, 2 red

    1 1 0 1 0

    1 1 0 0 1

    1 0 1 1 0

    1 0 1 0 1

    1 0 0 1 1

    0 1 1 1 0

    0 1 1 0 1

    0 1 0 1 1

    0 0 1 1 1

    Case 1:

    Your goal is to pick two of one color card in a row. (Your first pick is revealed to you before you pick the second cup)
    "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.

    Since the 26 distributions are equally likely, and your picks are random, absent any clues,

    all we need to do is see in how many of the outcomes the first two cups have the same color.

    The result is the same if we look at cups 2 and 5, or any other pair of cups.

    It turns out that in 12 cases the first two have the same color.

    Probability is 12/26 = 0.461

    Case 2:

    There is also an option in which you can pick one cup and the Host tells you what color paint chip is under it... but then you CANNOT pick that cup when you do your two picks.
    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

    Figure out the probability of winning (two same-color cards in a row) with and without that extra help option. Does it change?

    It makes sense that it doesn't matter about knowing what's under cup 1, since there is no way to make use of that knowledge in deciding which other two cups to pick.

    Since there is an unequal likelihood of any specific cup being red or green, your chances of picking two of the same color after eliminating a green cup [5/11 = .454] is different from your chances after eliminating a red cup [7/15 = .467]. But it is more likely [by a 15 to 11 margin -- look again at the distributions] that a red cup is eliminated. Weighting the two probabilities in this way leads to the expected average result of 12/26 = .461

  3. I have a question for you...

    Jonny has flipped: 101101 (6 flips so far)

    Albert has flipped: 0001000 (7 flips so far)

    What is the probability that Albert will get MORE 1's IN A ROW than Jonny?

    11/128

    Here's why:

    Johnny has two consecutive 1's, and, depending on his final 4 flips, he might end up with as many as 5.

    Albert has 3 remaining flips and could end up with 1, 2, or 3 consecutive 1's.

    For Albert to end up with MORE than Johnny,

    [1] Albert would need 3 and

    [2] Johnny would have to keep his total at 2.

    Probability of Albert getting 3 1's is 1/8.

    Johnny can flip 16 different outcomes.

    Of those 16, 4 have the first two as 1's; of the other 12, 1 outcome has the last three as 1's.

    Thus 5 of 16 possible outcomes give Johnny 3 or more consecutive 1's.

    The other 11 outcomes leave Johnny's total at 2.

    So the probability of [2] 11/16.

    Since the events are independent, the joint probability is the product:

    1/8 x 11/16 = 11/128.

  4. "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.

  5. 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.

  6. 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.

    Alex's result was remarkable in that the "goal" was consecutive Tails.

    But the others' claim that it would be difficult to repeat what they'd done was valid.

    The probability of repeating any of the results is 1 in 1024.

  7. 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.

    At the Royal Department of Weights and Measures, the King's

    scientists knew that you use a different system for weighing

    precious metals -- the Troy system -- from what you use for

    weighing say potatoes. Or for that matter, lead.

    That would be the Avoirdupois system.

    [hint: Helen = Helen of Troy]

    The systems differ enough for Alex to be able to deliver back

    to the King each morning only about 82% of the gold he made

    the previous day. At today's prices, Sir Alex was pocketing

    about $2000 each day. Not a bad return on his time.

    For the mathematically inclined, a pound is 16 oz in Avoirdupois,

    but only 12 oz in Troy weight. But Alex's cut wasn't 25%; it

    was less, because a Troy ounce [about 31 grams] is a little heavier

    than an Avoirdupois ounce [about 28 grams].

    So netting it all out, each day:

    Alex was given 453.6 grams of lead [1 pound Avoirdupois]

    Alex returned only 373.2 grams of gold [1 pound Troy]

    Alex kept about 80 grams of gold for himself.

  8. your body and consiounse. . . counscience. . . consience. . . whatever. . . both live forever, but are seperate, I guess.

    Who buys the soul?

    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.

  9. Ooo I know...

    Spoiler Is NO Fun... The answer is >>>

    The gold is the 1 lb lead which is painted with the color gold... Duh??

    If im wrong then i have no clue wat so ever....

    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?

  10. The answer is precisely 12/61 days.

    A = 1/2 (reservoirs/day)

    B = 1/3

    C = 1/4

    D = 4

    (A + B + C + D) * Time = 1 (Time is in days. A, B, C, and D are in reservoirs per day.)

    (1/2 + 1/3 + 1/4 + 4) * Time = 1

    (61 / 12) * Time = 1

    Time = 12 / 61 days (~ 4.72 hours)

    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.

  11. 5) latchstrings

    2) screeched, scrounged ???

    Answer time <!-- s:D --><!-- s:D -->

    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

  12. "Nobody goes to that restaurant; it's too crowded."

    That is a quote from the much-quoted, great Yogi Berra.

    Who is also quoted to have said,

    "I didn't say most of those things I said."

    So that possibly explains it.

    I also point out that George Carlin once noted that the stuff

    you spray on mold and mildew comes with the warning:

    Use this product only in well-ventilated places.

    To which George replied ...

    I don't know where your mildew grows, but ... :huh:

  13. Couldn't help ...

    replying to the notion that I don't have the ability to choose not

    to reply to the notion that I don't have the ability to choose not

    to reply to the notion that I don't have the ability to choose not

    to reply to the notion that I don't have the ability to choose not

    to reply to the notion that I don't have the ability to stop typing.

    But I must.

    Is that my destiny or did I choose freely to act as if it were?

    Ah, paradox, begone!

  14. So if he's not reselling the lead and making profit, and we are to assume he is making the lead into gold, then I propose:

    The means by which he is turning lead into gold preserves the number of atoms in the lead, so that none are lost in conversion. Therefore since a mole of gold is about 94% that of a mole of lead, then the alchemist is making about 6% on every pound of lead he's converting. Since the pound of lead would make about 1.06 pounds of gold, he gives up the pound of gold and keeps the 6%.

    You've got the right idea. But he makes a bigger cut -- about 20%.

    How?

  15. to brainiac:

    The simple answer is that there are two groups of people.

    .....

    I hope this clears up some of the confusion.

    Quite right. But this is an old result, first popularized by K Barth.

    BARTH'S DISTINCTION.

    There are two types of people:

    those who divide people into two types and those who do not.

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