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第 20 週 · 排列組合與概率 · 第 6 節

用排列組合算概率（＋本週 BOSS 總關卡）

把整週學的計數、排列、組合，全部接到「概率」上。古典概率 = 有利結果數 ÷ 總結果數 —— 上下兩個數都用排列或組合數出來。配滿圖例，最後一節含本週總關卡。

classical probabilityfavourable outcomes (count)total outcomes (count)= P
▶ 第一關： 古典概率 = 有利數 ÷ 總數
roll a die: P(even)123456favourable {2,4,6} = 3P = 3 / 6 = 1/2
52 cards: P(heart)13hearts52all cardsP = 13 / 52 = 1/4
▶ 第二關： 用組合算概率（不分先後）
5 boys + 3 girls, choose 35 boys3 girlsfavour: 5C3 = 10total: 8C3 = 56P(all boys) = 10/56 = 5/28
4 red + 2 blue, take 2, both red4 red2 bluefavour 4C2 = 6total 6C2 = 15P = 6/15 = 2/5
▶ 第三關： 相乘求有利數＋排列概率
5 boys 4 girls: choose 2B and 1Gboys: 5C2= 10×girls: 4C1= 4favour 10 × 4 = 40P = 40 / 9C3 = 40/84 = 10/21
arrange A B C: P(A first)A??fix A21favour 2! = 2, total 3! = 6P = 2/6 = 1/3
order matters?ask firstYES (rank/seat)use nPr / n!top & bottomNO (pick group)use nCrtop & bottom
⚔ BOSS： 本週總關卡（計數 / 排列 / 組合 / 概率）
W20 knowledge mapcountingnPr (ordered)nCr (unordered)P = favour / totaladd: A or Bmul: A and B
BOSS: 5 subtopics1 counting (and): 3 x 2 x 4 = 242 nPr: 7P3 = 7 x 6 x 5 = 2103 nCr+P: 5C3/9C3 = 10/84 = 5/424 add: 3/6+2/6-1/6 = 2/35 mul (no replace): 3/5 x 2/4 = 3/10