HMMT 二月 2000 · ADV 赛 · 第 9 题

HMMT February 2000 — ADV Round — Problem 9

The Cin innati Reals are pla ying the Houston Alphas in the last game of the Swirled Series. The Alphas are leading b y 1 run in the b ottom of the 9th (last) inning, and 1 2 the Reals are at bat. Ea h batter has a han e of hitting a single and a han e of 3 3 making an out. If the Reals hit 5 or more singles b efore they mak e 3 outs, they will win. If the Reals hit exa tly 4 singles b efore making 3 outs, they will tie the game and 3 send it in to extra innings, and they will ha v e a han e of ev en tually winning the game 5 (sin e they ha v e the added momen tum of oming from b ehind). If the Reals hit few er than 4 singles, they will LOSE! What is the probabilit y that the Alphas hold o the Reals and win, sending the pa k ed Alphadome in to a frenzy? Express the answ er as a fra tion.

解析

The probabilit y of the Reals hitting 0 singles is ( ) . The probabilit y of the Reals hitting 3 3 3 2 1 exa tly 1 single is ( ) , sin e there are 3 sp ots to put the t w o outs (the last sp ot 2 3 3 m ust b e an out, sin e the inning has to end on an out). The probabilit y of the Reals 3 3 4 2 1 hitting exa tly 2 singles is ( ) ( ) . The probabilit y of the Reals hitting exa tly 2 3 3 3 3 5 2 1 3 singles is ( ) ( ) . If an y of these happ en, the Alphas win righ t a w a y . Adding 2 3 3 656 these giv es us a han e of this happ ening. If exa tly 4 singles o ur (with probabilit y 729 3 4 6 2 1 2 ( ) ( ) ), then there is a han e that the Alphas win. The probabilit y of this 2 3 3 5 2 40 happ ening is . Th us, the total probabilit y of the Alphas winning is the sum of 5 729 656 16 224 these t w o probabilities, or + = . 729 729 243