HMMT 二月 2000 · ADV 赛 · 第 9 题
HMMT February 2000 — ADV Round — Problem 9
题目详情
英文原题
- 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 9 th (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 willwin. If the Reals hit exa tly 4 singles b efore making 3 outs, they will tie the game andsend it in to extra innings, and they will have a han e of ev en tually winning the game 3
(sin e they have the added momen tum of oming from b ehind). If the Reals hit few er 5
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 afra tion.
解析
英文解析
- The probabilit y of the Reals hitting 0 singles is ( ) . The probabilit y of the Reals hitting
2 1333
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 be an out, sin e the inning has to end on an out). The probabilit y of the Reals
3 3
2 14
hitting exa tly 2 singles is ( ) ( ) . The probabilit y of the Reals hitting exa tly
2 3 3
3 3
2 15
3 singles is ( ) ( ) . If an y of these happ en, the Alphas win righ t a w a y . Adding
2 3 3
these giv es us a han e of this happ ening. If exa tly 4 singles o ur (with probabilit y 656
3 4729
2 1 26
( ) ( ) ), 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