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Posts posted by jim

  1. You need to change to the standard definition of montone. Monotone increasing means each number is at least as large as the previous number. If each successive number is actually larger that is called strictly monotone. Decreasing monotone is defined in a similar way. Note 1,1,1,1,1,1,1,... would need my definition to have any montone sequence of length greater than one.

  2. As long as the process reaches completition within the year--or as long as the probability of A meeting B in a short interval of time does not depend on how many other people there are (a kind of Ideal Gas Law) your chances of survival are the same whether you chose to be a killer or a pacifist. That is because the only encounters that matter for you are when two killers meet, or when a killer meets you. Whether a pacifist dies when meeting you are not is irrelevant to your survival.

  3. Perhaps this will help clarify things. Suppose we find the average height of college men is greater than it was 60 years ago, the average height of college women is also greater, but the average height of college students is less than it was 60 years ago. Would this suggest that people are taller or shorter than they were 60 years ago.

  4. B for men, B for women, so obviously B is the better treatment regardless of gender--although this might not be statistically significant.

    But if you had only told me the overall numbers without a gender breakdown I would have picked A--but given the gender breakdown info I pick B.

  5. You don't need calculus. We just need to know the area of the part of the solid at least t / k cm away from an exposed surface. The radius decreases at the rate of t / k cm per second. The upper surface drops at the same rate, and so does the bottom surface except in the case where the bottom is not an exposed surface. As long as the height is at least 2R(or R if the bottom is not exposed) then the radius decreases linearly down to zero. Otherwise the height goes to zero while the radius approaches a positive limit.

    In your first point the limit is a point. With height 3R it approaches a limit that is a line of height R. If the height is R it takes half the time to disappear, and the limit is a 2D disc of radius R, etc.

  6. Any real number can be expressed by a sign, a finite number of binary digits before a binary point, and then a possibly iinfinite and definitely countable number of binary digits after the binary point. 0.1010010001... would be an example of an irrational number in binary where every run of 0"s countains one more 0 than the previous run and is followed by a single 1.

  7. EXTENSION 2--Same problem but now assume we add two balls red with probability r and blue with probability b. What about when we add three balls? This time it will we convienent to assume we start with 10 blue balls--but it is easy to generalize to k blue balls, just a little messy to write up the answer.

  8. --I flip the coin twice.

    HT is one outcome and TH is the other. If I get either HH or TT I repeat the procdure.

    -- I look for the first number in Pascal's triangle that is at least N. I select N possible outcomes with the same number of heads and tails specifying which correspond to j/N and which to (N-j)/N. In your puzzle j=1 but it could be any number less than N and relatively prime to N. A success is getting one of these outcomes.

    Following a failure we repeat the process with a difference. For our first try our best guess for the ratio of the likelyhood of getting head and tails was 1:1. Now it would be the maximum likelyhood estimate of (h+1):(t+1) where h is the total number of heads we have gotten up to this time and t is the total number of tails. This time we select a number from Pascal's triangle where the ratio of the likelyhood af a successful trial to the number of flips involved is as large as possible

  9. Suppose they each throw 100 times with 97 bullseyes for Alex while Davey gets 90 and Ian gets 80. What would the team of Davey and Ian score? Suppose Davey throws first and misses 10 times. Ian lets the pressure get to him and misses every time Davye does.lus 10 additional times. In this case the team scores 90. On the other hand, if Davey is good under pressure and makes all 10 of those throws while missing on 20 other occassions, then the team scores 100. If the correlation coefficent for Davey and Ian is zero, then the team scores 98 and they win. With a negative correlation they do even better, but with a large enoiugh positive correlation the bet will be a good onwe for Alex.

  10. Event Horizon--

    I think your proof is essentially the same as mine, but formulated in a different fashion.

    I basically used induction. We can do 3=3 and 4=3+1. Let m be the smallest number greater than 2 which we cannot do. It must be at least 5 so m-2 is at least 3 and as a smaller number greater than 2 we can do it. We adjust the solution for m-2 by increasing one of the odd numbers by two, replacing it with an odd nuimber not already included in the set of m-2. Since the solution for m-2 is not the null set, we can always do this unless the solution for m-2 is of the form U(j) which is all the allowable odd numbers from j upwards for some j less than or equal to 2n-1. But in this case we can replace j with 1 and 3 and j-2 which we can always do as long as j>5. If j=5 we get our sum m-2 = n^2 - 4 so m = n^2 -2 which is in fact the smallest non-solution except for 2. j=3 and j=1 give us m-2 = n^2 -1 and n^2 which in either case give us m> n^2 which we can also not do because they are too large.

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