+ ... + P

1. 2006

1. 2. 2006

1. 2. 3. 2006

1. 2. 3. 4. 2006

1. 2. 3. 4. 5. 2006

1. 2. 3. 4. 5. 6. 2006

Excel J. CardanoP. D. Fermat 2006

• 2006

• • (process) 2006

S = { , } S = { , , } 1 1 , 2 , 3 , 4 , 5 , 6 S = {1 , 2 , 3 , 4 , 5 , 6} S = { , } 2006

• 2006

• • S 2006

BBBGGBGGVenn 2006

BBBBBBGBGBGGBGGGBGBVenn J. VennGGG 2006

• 2006

B = {3}BS1 532 4 6 2006

A = {2 , 4 , 6}S1 532 4 6A 2006

• 2006

• • P rnn!=] n - r g!= n # ] n - 1g # g # ] n - r + 1g 2006

• • P rnn!=] n - r g!= n # ] n - 1g # g # ] n - r + 1g• C rnn!= r! ] n - r g!=n # ] n - 1g # g # ] n - r + 1gr! 2006

• P] E g1= N 2006

• P] E g1= N• P] E g = lim n " 3n] E gn 2006

• P] E g1= N• P] E g = lim n " 3• n] E gnP(E) = [ E ] 2006

• 0 ≤ P(Ei) ≤ 1 Ei 0 1 0 1 2006

• 0 ≤ P(Ei) ≤ 1 Ei 0 1 0 1• P(E1 E2 ... En) = P(E1) + P(E2) + ... + P(En)E1, E2, ..., En n E1, E2, ...,En E1 E2 En 2006

• 0 ≤ P(Ei) ≤ 1 Ei 0 1 0 1• P(E1 E2 ... En) = P(E1) + P(E2) + ... + P(En)E1, E2, ..., En n E1, E2, ...,En E1 E2 En • P(S) = 1 1 2006

☺•6C6 142 C6=5,245,786= 0.0000001906 2006

☺• • 6C 6 142C =6 5,245,7866C 11 C 5 1 # 642C =6 5,245,786= 0.0000001906= 0.000001144 2006

☺• • 6C 6 142C =6 5,245,7866C 11 C 5 1 # 642C =6 5,245,786= 0.0000001906= 0.000001144• 35 6C 1 C 5 35 # 642C =6 5,245,786= 0.00004003 2006

☺• • 6C 6 142C =6 5,245,7866C 11 C 5 1 # 642C =6 5,245,786= 0.0000001906= 0.000001144• 35 6C 1 C 5 35 # 642C =6 5,245,786= 0.00004003• 36 6C 2 C 4 630 # 1542C =6 5,245,786= 0.001801 2006

☺• • 6C 6 142C =6 5,245,7866C 11 C 5 1 # 642C =6 5,245,786= 0.0000001906= 0.000001144• 35 6C 1 C 5 35 # 642C =6 5,245,786= 0.00004003• • 36 6C 2 C 4 630 # 1542C =6 5,245,786= 0.00180136 6C 3 C 3 7,140 # 2042C =6 5,245,786= 0.02722 2006

• AP(A)A P(A) = ∑ P(Ei)Ei A 2006

... ...........................• AP(A)A P(A) = ∑ P(Ei)Ei A• A\B B1 B2 ... BcA1 A1 ∩ B1 A1 ∩ B2 ... A1 ∩ BcAr Ar ∩ B1 Ar ∩ B2 ... Ar ∩ Bc 2006

... ...........................• AP(A)A P(A) = ∑ P(Ei)Ei A• A\B B1 B2 ... BcA1 P(A1 ∩ B1) P(A1 ∩ B2) ... P(A1 ∩ Bc)Ar P(Ar ∩ B1 P(Ar ∩ B2) ... P(Ar ∩ Bc) 2006

ABB1 B2A1 320 80 400A2 14 36 50 334 116 450 2006

ABB1 B2A1 320 80 400A2 14 36 50 334 116 450P() = P(A1 ∩ B1) = 320/450 = 0.71ABB1 B2A1 P(A1 ∩ B1) = 0.71 P(A1 ∩ B2) = 0.18 P(A1) = 0.89A2 P(A2 ∩ B1) = 0.03 P(A2 ∩ B2) = 0.08 P(A2) = 0.11 P(B1) = 0.74 P(B2) = 0.26 1.00 2006

ABB1 B2A1 320 80 400A2 14 36 50 334 116 450P() = P(A2 ∩ B1) = 14/450 = 0.03ABB1 B2A1 P(A1 ∩ B1) = 0.71 P(A1 ∩ B2) = 0.18 P(A1) = 0.89A2 P(A2 ∩ B1) = 0.03 P(A2 ∩ B2) = 0.08 P(A2) = 0.11 P(B1) = 0.74 P(B2) = 0.26 1.00 2006

B1 P(A1 ∩ B1) = 0.71A1 B2 P(A1 ∩ B2) = 0.18A2 B1 P(A2 ∩ B1) = 0.03B2 P(A2 ∩ B2) = 0.08 2006

• AP(A)A P(A) = ∑ P(Ei)Ei A• • 2006

ABB1 B2A1 P(A1 ∩ B1) = 0.71 P(A1 ∩ B2) = 0.18 P(A1) = 0.89A2 P(A2 ∩ B1) = 0.03 P(A2 ∩ B2) = 0.08 P(A2) = 0.11 P(B1) = 0.74 P(B2) = 0.26 1.00 2006

ABB1 B2A1 P(A1 ∩ B1) = 0.71 P(A1 ∩ B2) = 0.18 P(A1) = 0.89A2 P(A2 ∩ B1) = 0.03 P(A2 ∩ B2) = 0.08 P(A2) = 0.11 P(B1) = 0.74 P(B2) = 0.26 1.00P() = P(B1) = P(A1 ∩ B1) + P(A2 ∩ B1)= 0.71 + 0.03 = 0.74 2006

• AP(A)A P(A) = ∑ P(Ei)Ei A• • • AB B A A 2006

ABB1 B2A1 P(A1 ∩ B1) = 0.71 P(A1 ∩ B2) = 0.18 P(A1) = 0.89A2 P(A2 ∩ B1) = 0.03 P(A2 ∩ B2) = 0.08 P(A2) = 0.11 P(B1) = 0.74 P(B2) = 0.26 1.00 2006

ABB1 B2A1 P(A1 ∩ B1) = 0.71 P(A1 ∩ B2) = 0.18 P(A1) = 0.89A2 P(A2 ∩ B1) = 0.03 P(A2 ∩ B2) = 0.08 P(A2) = 0.11 P(B1) = 0.74 P(B2) = 0.26 1.00P( | ) = P(B1 | A1) = = 0.80P(A1 ∩ B1)P(A1) 2006

• 2006

• AB AB P(A | B) = P(A) P(B | A) = P(B) P(A ∩ B) = P(A)P(B) 2006

24 10 34 12 16 28 36 26 622003/6/1~2003/8/31 2006

0.387 0.161 0.548 0.194 0.258 0.452 0.581 0.419 1.000 2006

0.387 0.161 0.548 0.194 0.258 0.452 0.581 0.419 1.000 P(TPU ∩ NYU) = 0.706 P(TPD ∩ NYU) = 0.294 P(TPU ∩ NYD) = 0.429 P(TPD ∩ NYD) = 0.571 2006

• • 2006

• • • 2006

B1B1 A 1 175 225 400A2 100 200 300 275 425 700 2006

B1B1 A 1 175 225 400A2 100 200 300 275 425 700175P^B 1 |A 1 h = = 44%400100P^B 1 |A 2 h = = 33%300 2006

150 50 25 50 150 50 75 150 300 100 100 200 2006

150 50 25 50 150 50 75 150 300 100 100 200P( | ∩ ) = 150/300 = 1/2P( | ∩ ) = 50/100 = 1/2P( | ∩ ) = 25/100 = 1/4P( | ∩ ) = 50/200 = 1/4 2006

• P(A B) = P(A) + P(B) − P(A ∩ B) A B P(A B) = P(A) + P(B)• P(A ∩ B) = P(B)P(A | B) AB P(A ∩ B) = P(A)P(B) 2006

• A1, ... , Ai B r/i = 1] gP] Bg = P B + A iP] B + A i g = P] A i g $ P^B |A i hrP] Bg = / P] B + A i g = P A ii = 1r/i = 1] g $ P^B |A ih 2006

A1, ... , Ai B P(Ai) P(B | Ai) B Ai P(Ai | B)P] B + AP^ A i |B h =i g P] B + AP ] B g =i gP ] B + A1 g + P] B + A 2 g + g + P] B + A r gP] A=i gP(B |A i )P ] A1 gP(B |A 1 ) + P] A 2 gP(B |A 2 ) + g + P] A r gP(B |A r ) 2006

P^B |A 1 h = 0.9P] A 1 + Bg= P] A 1 g $ P^B |A 1= 0.6 # 0.9 = 0.54hP^ A 1 |B h = 0.82P] A 1 g = 0.6P^B |A 1 h = 0.1P^A 1 + B= P] A 1 g $ P^B |A 1 h= 0.6 # 0.1 = 0.06hP^ A 1 |B h = 0.18P] A 2 g = 0.4P^B |A 2 h = 0.3P] A 2 + Bg= P] A 2 g $ P^B |A 2= 0.4 # 0.3 = 0.12hP^ A 2 |B h = 0.18P^B |A 2 h = 0.7P^A 2 + B h= P] A 2 g $ P^B |A 2 h= 0.4 # 0.7 = 0.28P^ A 2 |B h = 0.82 2006

2006

2006

2006

2006