Question

（本小題滿分12分）

在科技館里，小亮看見一臺名為帕斯卡三角的儀器，如圖所示，當一實心小球從入口落下，它在依次碰到每層菱形擋塊時，會等可能地向左或向右落下．

（1）試問小球通過第二層位置的概率是多少？

（2）請用學過的數(shù)學方法模擬試驗，并具體說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

【答案】

方法1：①實心小球在碰到菱形擋塊時向左或向右下落是等可能性的經(jīng)過一個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

畫樹狀圖可知，落到點位置的概率為．·········································· 4分

②同理可畫樹狀圖得，落到點位置的概率為．·································· 8分

③同理可畫樹狀圖得，落到點位置的概率為．······························ 12分

（注：①中畫圖1分，算出概率2分．②、③中畫圖2分，算出概率2分．）

方法2：（1）實心小球碰到每個菱形擋塊時向左或向右是等可能性的，因此小球下落到的可能性會有以下的途徑｛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，到達第三層位置會有以下途徑｛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，到達第四層位置會有｛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)過每條路徑的可能性相同．

 

由概率定義易得，（其中畫圖2分，算出概率2分）············· 4分

，（其中畫圖2分，算出概率2分）·································· 8分

．（其中畫圖2分，算出概率2分）····················· 12分

（注：其它方案正確，可參照上述方案評分�。�

【解析】略