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I'm thinking about starting a maths competition. One which any one can participate and any methods can be used in the competition including astrology, palmistry or numerology.
The objective is to estimate the true probability of obtaining "Head" on a bias coin. The true probability of obtaining head with the bias coin can be expected range from 0.1 to 0.9 , in fact the coin toss is simulated by using a table of random numbers.
The procedure of the competition is as follows. The outcome of the coin toss would be annouced and each competitor would display their best estimate using all the knowledge they have obtained so far.
The full competition would ran for 40 rounds (thus involving 40 coin toss).
The winner would be the participant which has the lowest sum of error square.
A dummy run (of 3 rounds) is as follows: Assume that there are 3 competitors A, B and C.
The competition has ended and the judge announces the true probability of "Head" for the coin is 0.55 then the scores are calculated.
Coin Toss | Actual value | A's estimate | A's error square | B's estimate | B's error square | C's estimate | C's error square |
---|---|---|---|---|---|---|---|
Head | 0.55 | 1.0 | 0.2025 | 0.6 | 0.0025 | 0.5 | 0.0025 |
Tail | 0.55 | 0.0 | 0.3025 | 0.4 | 0.0225 | 0.4 | 0.0225 |
Head | 0.55 | 0.7 | 0.0225 | 0.7 | 0.0225 | 0.5 | 0.0025 |
Total | 0.5275 | Total | 0.0475 | Total | 0.0275 |
The judge declares C the winner for having the smallest total of error square.
My question is this. If you want to win this competition, what is the best strategy and method for calculating the probability of "Head" in order to obtain the lowest possible error square score. Ohanian 01:33, 29 April 2007 (UTC)
(* Mathematica Input Format *)
- f0[x_] := 5/4;
- TEXT: Judge calls "Head"
- f1[x_] := x*(f0[x]/Integrate[s*f0[s], {s, 0.1, 0.9}]);
- Plot[f1[x], {x, 0.1, 0.9}];
- errorsquare1[g_] := Integrate[(x - g)^2*f1[x],{x, 0.1, 0.9}]
- OUT[4]= 0.41 - 1.213*g + 1.*g^2
- Plot[errorsquare1[g], {g, 0.1, 0.9}];
- FindMinimum[errorsquare1[g], {g, 0.501}]
- OUT[6]= {0.0419556,{g->0.6067}}
- TEXT: Judge calls "Tail"
- f2[x_] := (1 - x)*(f1[x]/Integrate[(1-s)*f1[s],{s, 0.1, 0.9}]);
- Plot[f2[x], {x, 0.1, 0.9}]
- errorsquare2[g_] := Integrate[(x - g)^2*f2[x],{x, 0.1, 0.9}]
- Plot[errorsquare2[g], {g, 0.1, 0.9}];
- FindMinimum[errorsquare2[g], {g, 0.501}]
- OUT[11]= {0.0271, {g -> 0.5}}
- TEXT: Judge calls "Head"
- f3[x_] := x*(f2[x]/Integrate[s*f2[s], {s, 0.1, 0.9}]);
- Plot[f3[x], {x, 0.1, 0.9}];
- errorsquare3[g_] := Integrate[(x - g)^2*f3[x],{x, 0.1, 0.9}]
- Plot[errorsquare3[g], {g, 0.1, 0.9}];
- FindMinimum[errorsquare3[g], {g, 0.501}]
OUT[16]= {0.0348,{g->0.5835}}
Coin Toss | Actual value | Lambiam's estimate | Lambiam's error square |
---|---|---|---|
Head | 0.55 | 0.6067 | 0.0032 |
Tail | 0.55 | 0.50 | 0.0025 |
Head | 0.55 | 0.5835 | 0.0011 |
Total | 0.0068 |
I was wondering if there is a math property that allows you to move numbers around. ei 1+9*8+3. could you switch the 9 and the 8 around? —The preceding unsigned comment was added by 75.162.155.149 ( talk) 02:03, 29 April 2007 (UTC).
Hey all, back again with another question from a previous qualifying examination that I can't quite seem to figure out. Here's the statement:
Let be continuous, with f(0) = 0 and exists. Show that for all .
Progress. Just so you don't think I'm a complete bum, let me tell you what I have right now. Since f is differentiable at 0, we know that exists. This suggests doing something like this:
and then note that since p < 2, . I feel like there should be some kind of use of Hoelder's inequality, but I can't quite figure out which two functions I would want to use. Hints would be greatly appreciated. Thanks! – King Bee ( τ • γ) 17:45, 29 April 2007 (UTC)
Hi. I've been working through Number Theory by George Andrews, and doing each exercise, but I'm stuck on one where I think the text is in error. Either that, or I'm making a dumb mistake, because when I apply the hint given, it doesn't work. I've searched for a published errata for the book, but can't seem to find anything. The problem is #5 in section 8.2, on page 111.
The problem is connected with the previous one, which establishes an inequality following from the assumption that there is no prime number between and . In #5, we're asked to show that the inequality in #4 is impossible, given a couple of assumptions, which are both inequalities.
The equality that needs to be shown impossible (for sufficiently large n) is: , where denotes the product of all primes not exceeding .
The two inequalities that are supposed to contradict that one are and .
The only sensible way I can see to combine those inequalities is just to line them up, and deduce that , but that doesn't seem to contradict anything, since you've got on the left, and something bigger than on the right.
Am I missing something, or is there a typo in the book, or what? Has somebody encountered this problem before, and made sense of it? - GTBacchus( talk) 19:26, 29 April 2007 (UTC)
Mathematics desk | ||
---|---|---|
< April 28 | << Mar | April | May >> | April 30 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
I'm thinking about starting a maths competition. One which any one can participate and any methods can be used in the competition including astrology, palmistry or numerology.
The objective is to estimate the true probability of obtaining "Head" on a bias coin. The true probability of obtaining head with the bias coin can be expected range from 0.1 to 0.9 , in fact the coin toss is simulated by using a table of random numbers.
The procedure of the competition is as follows. The outcome of the coin toss would be annouced and each competitor would display their best estimate using all the knowledge they have obtained so far.
The full competition would ran for 40 rounds (thus involving 40 coin toss).
The winner would be the participant which has the lowest sum of error square.
A dummy run (of 3 rounds) is as follows: Assume that there are 3 competitors A, B and C.
The competition has ended and the judge announces the true probability of "Head" for the coin is 0.55 then the scores are calculated.
Coin Toss | Actual value | A's estimate | A's error square | B's estimate | B's error square | C's estimate | C's error square |
---|---|---|---|---|---|---|---|
Head | 0.55 | 1.0 | 0.2025 | 0.6 | 0.0025 | 0.5 | 0.0025 |
Tail | 0.55 | 0.0 | 0.3025 | 0.4 | 0.0225 | 0.4 | 0.0225 |
Head | 0.55 | 0.7 | 0.0225 | 0.7 | 0.0225 | 0.5 | 0.0025 |
Total | 0.5275 | Total | 0.0475 | Total | 0.0275 |
The judge declares C the winner for having the smallest total of error square.
My question is this. If you want to win this competition, what is the best strategy and method for calculating the probability of "Head" in order to obtain the lowest possible error square score. Ohanian 01:33, 29 April 2007 (UTC)
(* Mathematica Input Format *)
- f0[x_] := 5/4;
- TEXT: Judge calls "Head"
- f1[x_] := x*(f0[x]/Integrate[s*f0[s], {s, 0.1, 0.9}]);
- Plot[f1[x], {x, 0.1, 0.9}];
- errorsquare1[g_] := Integrate[(x - g)^2*f1[x],{x, 0.1, 0.9}]
- OUT[4]= 0.41 - 1.213*g + 1.*g^2
- Plot[errorsquare1[g], {g, 0.1, 0.9}];
- FindMinimum[errorsquare1[g], {g, 0.501}]
- OUT[6]= {0.0419556,{g->0.6067}}
- TEXT: Judge calls "Tail"
- f2[x_] := (1 - x)*(f1[x]/Integrate[(1-s)*f1[s],{s, 0.1, 0.9}]);
- Plot[f2[x], {x, 0.1, 0.9}]
- errorsquare2[g_] := Integrate[(x - g)^2*f2[x],{x, 0.1, 0.9}]
- Plot[errorsquare2[g], {g, 0.1, 0.9}];
- FindMinimum[errorsquare2[g], {g, 0.501}]
- OUT[11]= {0.0271, {g -> 0.5}}
- TEXT: Judge calls "Head"
- f3[x_] := x*(f2[x]/Integrate[s*f2[s], {s, 0.1, 0.9}]);
- Plot[f3[x], {x, 0.1, 0.9}];
- errorsquare3[g_] := Integrate[(x - g)^2*f3[x],{x, 0.1, 0.9}]
- Plot[errorsquare3[g], {g, 0.1, 0.9}];
- FindMinimum[errorsquare3[g], {g, 0.501}]
OUT[16]= {0.0348,{g->0.5835}}
Coin Toss | Actual value | Lambiam's estimate | Lambiam's error square |
---|---|---|---|
Head | 0.55 | 0.6067 | 0.0032 |
Tail | 0.55 | 0.50 | 0.0025 |
Head | 0.55 | 0.5835 | 0.0011 |
Total | 0.0068 |
I was wondering if there is a math property that allows you to move numbers around. ei 1+9*8+3. could you switch the 9 and the 8 around? —The preceding unsigned comment was added by 75.162.155.149 ( talk) 02:03, 29 April 2007 (UTC).
Hey all, back again with another question from a previous qualifying examination that I can't quite seem to figure out. Here's the statement:
Let be continuous, with f(0) = 0 and exists. Show that for all .
Progress. Just so you don't think I'm a complete bum, let me tell you what I have right now. Since f is differentiable at 0, we know that exists. This suggests doing something like this:
and then note that since p < 2, . I feel like there should be some kind of use of Hoelder's inequality, but I can't quite figure out which two functions I would want to use. Hints would be greatly appreciated. Thanks! – King Bee ( τ • γ) 17:45, 29 April 2007 (UTC)
Hi. I've been working through Number Theory by George Andrews, and doing each exercise, but I'm stuck on one where I think the text is in error. Either that, or I'm making a dumb mistake, because when I apply the hint given, it doesn't work. I've searched for a published errata for the book, but can't seem to find anything. The problem is #5 in section 8.2, on page 111.
The problem is connected with the previous one, which establishes an inequality following from the assumption that there is no prime number between and . In #5, we're asked to show that the inequality in #4 is impossible, given a couple of assumptions, which are both inequalities.
The equality that needs to be shown impossible (for sufficiently large n) is: , where denotes the product of all primes not exceeding .
The two inequalities that are supposed to contradict that one are and .
The only sensible way I can see to combine those inequalities is just to line them up, and deduce that , but that doesn't seem to contradict anything, since you've got on the left, and something bigger than on the right.
Am I missing something, or is there a typo in the book, or what? Has somebody encountered this problem before, and made sense of it? - GTBacchus( talk) 19:26, 29 April 2007 (UTC)