Question

4£®ÓûùÒòÐÍΪAaµÄСÂó·Ö±ð½øÐТñËæ»ú½»Åä¡¢¢òËæ»ú½»Åä²¢Öð´úÌÔÌ­ÒþÐÔ¸öÌå¡¢¢óÁ¬Ðø×Ô½»²¢Öð´úÌÔÌ­ÒþÐÔ¸öÌå¡¢¢ôÁ¬Ðø×Ô½»£¬¸ù¾Ý¸÷´úAa»ùÒòÐÍƵÂÊ»æÖÆÇúÏßÈçͼ£®Çë¾Ýͼ»Ø´ðÏÂÁÐÎÊÌ⣺
£¨1£©Ëæ»ú½»ÅäµÄÏàÓ¦ÇúÏßÊÇ¢ñ£»ÇúÏߢòµÄF₃ÖÐAa»ùÒòÐÍƵÂÊÊÇ$\frac{2}{5}$£¨Ó÷ÖÊý±íʾ£©£®
£¨2£©ÇúÏߢóµÄF_nÖÐAa»ùÒòÐÍƵÂÊΪ$\frac{2}{1{+2}^{n}}$£¨Ó÷ÖÊý±íʾ£©£®
£¨3£©Ëĸö²»Í¬Ìõ¼þÏ·¢ÉúÁ˽ø»¯µÄÖÖȺÊÇÇúÏߢòºÍ¢ó£®

Answer 1

·ÖÎö Á¬Ðø×Ô½»ºÍËæ»ú½»ÅäµÄF₁µÄAaµÄ»ùÒòƵÂʶ¼ÊÇ$\frac{1}{2}$£¬ËùÒÔIºÍIV·ûºÏ£¬µ«ÊÇÁ¬Ðø×Ô½»µÄ½á¹ûÊÇ´¿ºÏ×ÓËùÕ¼µÄ±ÈÀýÔ½À´Ô½´ó£¬ÔÓºÏ×ÓËùÕ¼µÄ±ÈÀýÔ½À´Ô½Ð¡£¬ËùÒÔIÊÇËæ»ú½»ÅäµÄ½á¹û£¬IVÊÇ×Ô½»µÄ½á¹û£®ÇúÏߢòºÍ¢óÔÚF₁ÔÓºÏ×ÓAaËùÕ¼µÄ±ÈÀýÏàͬ£¬ÕâÊÇÓÉÓÚ×Ô½»ºÍËæ»ú½»ÅäµÄ½á¹ûÊÇÒ»ÑùµÄ£¬¼´F₁µÄ»ùÒòÐͼ°Æä±ÈÀýΪ£º£¨$\frac{1}{4}$AA+$\frac{1}{2}$Aa+$\frac{1}{4}$aa£©£¬ÌÔÌ­µôÆäÖеÄaa¸öÌåºó£¬AaµÄ±ÈÀý±äΪ$\frac{2}{3}$£¬AAµÄ±ÈÀýΪ$\frac{1}{3}$£¬Èç¹û×Ô½»ÔòÆäºó´úÊÇ$\frac{1}{3}$AA+$\frac{2}{3}$Aa£¨$\frac{1}{4}$AA+$\frac{1}{2}$Aa+$\frac{1}{4}$aa£©£¬ÌÔÌ­µôaaÒԺ󣬵õ½µÄºó´úF₃ÊÇ$\frac{3}{5}$AA+$\frac{2}{5}$Aa£¬AaËùÕ¼µÄ±ÈÀýÊÇ0.4£®Èç¹ûËæ»ú½»Å䣬¸ù¾ÝÒÅ´«Æ½ºâ¶¨ÂÉ£¨$\frac{2}{3}$A+$\frac{1}{3}$a£©²£¬ºó´úÊÇ£¨$\frac{4}{9}$AA+$\frac{4}{9}$Aa+$\frac{1}{9}$aa£©£¬ÌÔÌ­µôaa£¬ÔòF₃ÊÇ$\frac{1}{2}$AA+$\frac{1}{2}$Aa£¬ËùÒÔ´ÓÕâÀï¿´ÒÔ¿´³öÇúÏߢòÊÇËæ»ú½»Åä²¢ÌÔÌ­aaµÄÇúÏߣ¬ÇúÏߢóÊÇ×Ô½»²¢ÌÔÌ­aaµÄÇúÏߣ®

½â´ð ½â£º£¨1£©ÓÉÌâÒâ¿ÉÖª£¬ÓûùÒòÐÍΪAaµÄСÂóËæ»ú½»Å䣬ºó´úÖÐAaµÄ»ùÒòÐÍƵÂÊΪ2¡Á$\frac{1}{2}¡Á\frac{1}{2}$=$\frac{1}{2}$£¬ÇÒ²»ÂÛ½»Åä¶àÉÙ´ú£¬AaµÄ»ùÒòÐÍƵÂʲ»±ä£¬ËùÒÔËæ»ú½»ÅäµÄÏàÓ¦ÇúÏßÊÇ¢ñ£®ÇúÏߢòÊÇËæ»ú½»Åä²¢ÌÔÌ­aaµÄÇúÏߣ¬F₃Ëæ»ú½»ÅäÒÔºó£¨$\frac{3}{4}$A+$\frac{1}{4}$a£©²£¬¼´$\frac{9}{16}$AA+$\frac{6}{16}$Aa+$\frac{1}{16}$aa£¬ÌÔÌ­µôaaÒÔºóΪ$\frac{3}{5}$AA+$\frac{2}{5}$Aa£¬Aa»ùÒòÐÍƵÂÊΪ$\frac{2}{5}$£¬ËùÒÔF₃ÖÐAaµÄ»ùÒòÐÍƵÂÊΪ$\frac{2}{5}$£®
£¨2£©ÇúÏߢóÊÇ×Ô½»²¢ÌÔÌ­aaµÄÇúÏߣ¬F_nÖÐAaµÄ±ÈÀýΪ$\frac{1}{{2}^{n}}$£¬AAµÄ±ÈÀýΪ$\frac{1}{2}£¨$1-$\frac{1}{{2}^{n}}$£©£¬Òò´ËF_nÖÐAa»ùÒòÐÍƵÂÊΪ$\frac{Aa}{Aa+AA}$=$\frac{\frac{1}{{2}^{n}}}{\frac{1}{{2}^{n}}+\frac{1}{2}¡Á£¨1-\frac{1}{{2}^{n}}£©}$=$\frac{2}{1{+2}^{n}}$£®
£¨3£©Á¬Ðø×Ô½»²¢Öð´úÌÔÌ­ÒþÐÔ¸öÌå¡¢Ëæ»ú½»Åä²¢Öð´úÌÔÌ­ÒþÐÔ¸öÌåÓÉÓÚÈ˹¤Ñ¡Ôñ×÷Óã¬Ê¹A»ùÒòƵÂÊÉý¸ß£¬a»ùÒòƵÂÊϽµ£¬ÉúÎï·¢ÉúÁ˽ø»¯£®Òò´ËËĸö²»Í¬Ìõ¼þÏ·¢ÉúÁ˽ø»¯µÄÖÖȺÊÇÇúÏߢòºÍ¢ó£®
¹Ê´ð°¸Îª£º
£¨1£©¢ñ$\frac{2}{5}$
£¨2£©$\frac{2}{1{+2}^{n}}$
£¨3£©¢òºÍ¢ó

µãÆÀ ±¾Ì⿼²é»ùÒò·ÖÀ붨Âɼ°Ó¦ÓᢻùÒòƵÂʵÄÏà¹Ø¼ÆË㣬ҪÇó¿¼ÉúÕÆÎÕ»ùÒò·ÖÀ붨ÂɵÄʵÖʺÍÒÅ´«Æ½ºâ¶¨ÂÉ£¬ÄÜÅжÏͼÖÐËÄÌõÇúÏß·Ö±ð´ú±íÌâ¸ÉÖеÄÄÄÖÖÇé¿ö£¬ÔÙ½áºÏÑ¡ÏÔËÓûùÒò·ÖÀ붨ÂɺͻùÒòƵÂʽøÐÐÏà¹ØµÄ¼ÆË㣬ÓÐÒ»¶¨ÄѶȣ®