Question

（本小題滿分12分）
在科技館里，小亮看見(jiàn)一臺(tái)名為帕斯卡三角的儀器，如圖所示，當(dāng)一實(shí)心小球從入口落下，它在依次碰到每層菱形擋塊時(shí)，會(huì)等可能地向左或向右落下．
（1）試問(wèn)小球通過(guò)第二層位置的概率是多少？
（2）請(qǐng)用學(xué)過(guò)的數(shù)學(xué)方法模擬試驗(yàn)，并具體說(shuō)明小球下落到第三層位置和第四層
位置處的概率各是多少？
 

Answer 1

方法1：①實(shí)心小球在碰到菱形擋塊時(shí)向左或向右下落是等可能性的經(jīng)過(guò)一個(gè)菱形擋塊后向左或向右下落的概率各是原概率的一半．············································································ 1分
畫(huà)樹(shù)狀圖可知，落到點(diǎn)位置的概率為．·········································· 4分
②同理可畫(huà)樹(shù)狀圖得，落到點(diǎn)位置的概率為．·································· 8分
③同理可畫(huà)樹(shù)狀圖得，落到點(diǎn)位置的概率為．······························ 12分
（注：①中畫(huà)圖1分，算出概率2分．②、③中畫(huà)圖2分，算出概率2分．）

方法2：（1）實(shí)心小球碰到每個(gè)菱形擋塊時(shí)向左或向右是等可能性的，因此小球下落到的可能性會(huì)有以下的途徑｛左右，右左｝兩種情況，···················································································· 1分
而下落到第二層，共｛左左，左右，右左，右右｝四種情況·································· 2分
由概率定義得······································································· 4分
（2）同理，到達(dá)第三層位置會(huì)有以下途徑｛左右右，右左右，右右左｝三種情況
·········································································································· 5分
而下落到第三層共有｛左左左，左左右，左右左，左右右，右左左，右左右，右右左，右右右｝八種情況       6分
由概率定義得············································································· 8分
（3）同理，到達(dá)第四層位置會(huì)有｛左左左右，左左右左，左右左左，右左左左｝四種情況     9分
而下落到第四層共有｛左左左左，左左左右，左左右左，左右左左，右左左左，左右左右，左右右左，左左右右，右左左右，右左右左，右右左左，右右右左，右右左右，右左右右，左右右右，右右右右｝共16情況  10分
由概率定義得····································································· 12分
方法3：本題也可用賈憲三角方法，先算出小球下落路徑條數(shù)，如下圖．由題意知：小球經(jīng)過(guò)每條路徑的可能性相同．
由概率定義易得，（其中畫(huà)圖2分，算出概率2分）············· 4分
，（其中畫(huà)圖2分，算出概率2分）·································· 8分
．（其中畫(huà)圖2分，算出概率2分）····················· 12分
（注：其它方案正確，可參照上述方案評(píng)分！）

解析