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HF: 发表于 2012-3-19 12:55:57

硬币题

<br /="/"/>抛一硬币 （正反概率各1/2），你可以决定何时停止。一旦停止，你得到的奖励为K/N， 这里，N是停止时总抛币数，K是其中正面的次数。<br /="/"/>问：如何决定何时停止使得你的平均所得最多？<br /="/"/><div></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div>

pf_123456 发表于 2012-3-21 01:40:30

whenever you get more than half, then stop

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">1.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">If 1^st = 1, then stop. If 1^st = 0 then continual.</span><span style="display:none;">www.ddhw.com</span></p><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">2.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">If 2^nd = 0 then continual; if 2^nd = 1 then continual too, </span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">because 50% chance to get 2/3.</span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin"></span><span style="display:none;">www.ddhw.com</span></p><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">3.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">A: If 2^nd = 0 and 3^rd = 0, </span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">then continual; B: if 2^nd = 0 and 3^rd = 1 or 2^nd = 1 and 3^rd = 0 then continual because the expected value = 3/8 > 1/3. <span style="mso-spacerun: yes"> </span>C: If 2^nd = 1 and 3^rd = 1 then stop because the expected value = 5/8 < 2/3.</span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin"></span><span style="display:none;">www.ddhw.com</span></p><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">4.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">Keep doing on this way until you get more than ½.</span></p></div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

HF: 发表于 2012-3-21 04:32:56

回复：whenever you get more than half, then stop

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div><font face="Calibri" size="2"></font> </div><div><font face="Calibri" size="2">...if 2^nd = 0 and 3^rd = 1 or 2^nd = 1 and 3^rd = 0 then continual because the expected value = 3/8 > 1/3...</font></div><div><font face="Calibri" size="2"></font> </div><div><font face="Calibri" size="2">why expected value = 3/8 here?</font><span style="display:none;">www.ddhw.com</span></div><br /="/"/><table width="100%"><tr><td colspan="10"></td><td width="100%"><i>原贴：</i><hr /="/"/><span>文章来源: pf_123456 于 2012-3-20 17:40:30 (北京时间: 2012-3-21 5:40:30)<br /="/"/>标题：<b>whenever you get more than half, then stop</b><br /="/"/><table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">1.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">If 1^st = 1, then stop. If 1^st = 0 then continual.</span><span style="display:none;">www.ddhw.com</span></p><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">2.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">If 2^nd = 0 then continual; if 2^nd = 1 then continual too, </span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">because 50% chance to get 2/3.</span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin"></span><span style="display:none;">www.ddhw.com</span></p><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">3.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">A: If 2^nd = 0 and 3^rd = 0, </span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 10pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">then continual; B: if 2^nd = 0 and 3^rd = 1 or 2^nd = 1 and 3^rd = 0 then continual because the expected value = 3/8 > 1/3. <span style="mso-spacerun: yes"> </span>C: If 2^nd = 1 and 3^rd = 1 then stop because the expected value = 5/8 < 2/3.</span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin"></span></p><p style="TEXT-INDENT: -0.25in; MARGIN: 0in 0in 0pt 0.5in; mso-list: l0 level1 lfo1"><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin"><span style="mso-list: Ignore">4.<span style="FONT: 7pt 'Times New Roman'">       </span></span></span><span style="FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin">Keep doing on this way until you get more than ½.</span></p></div></td></tr></table><hr /="/"/></span></td></tr></table><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

fov22 发表于 2012-3-21 13:18:20

回复：whenever you get more than half, then stop

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>赞同pf的结论，</div><div>但是理由粗略的想了一下，要是奖励没到1/2 继续玩，根据大数定理（strong law of large numbers）， 最后出现超过或者等于1/2的概率为1.</div><div>要是在某刻的奖励m/n>1/2, 那么接下来的期望是m+K/n+2K 对于任何接下来的K次投币， 应该可以观察出这个期望总是小于m/n 的，所以应该停止</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

HF: 发表于 2012-3-22 05:48:31

回复：回复：whenever you get more than half, then stop

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>“要是在某刻的奖励m/n>1/2, 那么接下来的期望是m+K/n+2K 对于任何接下来的K次投币”</div><div>没这么简单。因为“接下来”，你还可以选择何时停止，而不是固定再投K次币。</div><br /="/"/><table width="100%"><tr><td colspan="10"></td><td width="100%"><i>原贴：</i><hr /="/"/><span>文章来源: fov22<span>®</span> 于 2012-3-21 5:18:20 (北京时间: 2012-3-21 17:18:20)<br /="/"/>标题：<b>回复：whenever you get more than half, then stop</b><br /="/"/><table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>赞同pf的结论，</div><div>但是理由粗略的想了一下，要是奖励没到1/2 继续玩，根据大数定理（strong law of large numbers）， 最后出现超过或者等于1/2的概率为1.<span style="display:none;">www.ddhw.com</span></div><div>要是在某刻的奖励m/n>1/2, 那么接下来的期望是m+K/n+2K 对于任何接下来的K次投币， 应该可以观察出这个期望总是小于m/n 的，所以应该停止</div></td></tr></table><hr /="/"/></span></td></tr></table><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

biomence 发表于 2012-3-25 16:44:47

这题是想钓出5毛党来吗？[:-Q]

<br /="/"/>  这题是想钓出5毛党来吗？ <img /="/" border="0" src="/img/23.gif"></img><br /="/"/><br /="/"/><br /="/"/><br /="/"/><br /="/"/><div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div>

fov22 发表于 2012-3-31 04:11:15

回复：回复：回复：whenever you get more than half, then sto

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>你有答案么说出来听听。</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

学生 发表于 2012-4-10 06:01:07

回复：硬币题

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>这是一个涉及随机游动的问题。只能说“在第一次出现正面时就打住”（得到的奖励为1）；但当第一次出现反面时，就没有策略能保证达到“<strong>最</strong>优”（就象开区间(0, 1)中没有<strong>最</strong>大的数那样）。</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

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