> 0.5

0.5

P(A|B)

\neq P(A)

P(A) = 0.9

P(A|B) \neq P(A)

P(A|B)

P(A) = \frac{5}{36}

(1 , 1) | (1 , 2) | (1 , 3) | (1 , 4) | (1 , 5) | (1 , 6) |
---|---|---|---|---|---|

(2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |

(3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |

(4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |

(5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |

(6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |

P(A|B) = \frac{1}{6}

(1 , 1) | (1 , 2) | (1 , 3) | (1 , 4) | (1 , 5) | (1 , 6) |
---|---|---|---|---|---|

(2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |

(3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |

(4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |

(5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |

(6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |

P(B) = \frac{6}{36}

\Omega

(6, 2)

(1, 1)~(1,2)~(1,3)~(1,4)~(1,5)~(1,6)

(4, 4)

(3, 5)

(5, 3)

(4, 1)~~(4,2)~~(4,3)~~(4,5)~~(4,6)

(2, 6)

(2, 1)~(2,2)~(2,3)~(2,4)~(2,5)

(3, 1)~(3,2)~(3,3)~(3,4)~(3,6)

(5, 1)~(5,2)~(5,4)~(5,5)~(5,6)

(6, 1)~(6,3)~(6,4)~(6,5)~(6,6)

A

B

A \cap B

P(A\cap B) = \frac{1}{36}

P(A|B) = \frac{P(A\cap B)}{P(B)} = \frac{\frac{1}{36}}{\frac{6}{36}} = \frac{1}{6}

P(A|B)

P(A|B) = \frac{P(A\cap B)}{P(B)}

P(A) = \frac{20}{90} = \frac{2}{9}

P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{20}{90}}{\frac{65}{90}} = \frac{4}{13}

A

B

P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.6} = \frac{1}{3}

A

B

\Omega

P(A|B) = \frac{P(A\cap B)}{P(B)} \geq 0

P(\Omega|A)= \frac{P(\Omega \cap B)}{P(B)} = \frac{P(B)}{P(B)} = 1

P(A_1 \cup A_2|B) = \frac{P((A_1 \cup A_2) \cap B)}{P(B)}

= \frac{P(A_1 \cap B)}{P(B)} + \frac{P(A_2 \cap B)}{P(B)}

= P(A_1|B) + P(A_2|B)

B

A_1

\Omega

A_2

= \frac{P((A_1 \cap B) \cup (A_2 \cap B)}{P(B)}

P(A|B) = \frac{P(A\cap B)}{P(B)}

P(B|A) = \frac{P(B\cap A)}{P(A)}

\therefore P(A\cap B) = P(A|B)\cdot P(B)

\therefore P(B\cap A) = P(B|A)\cdot P(A)

\therefore P(A\cap B) = P(A|B)\cdot P(B) = P(B|A)\cdot P(A)

+

B

\Omega

A

A\cap B

A\cap B^\mathsf{c}

A^\mathsf{c}\cap B

A^\mathsf{c}\cap B^\mathsf{c}

+

B

\Omega

A

P(A) = 0.1

P(B^\mathsf{c}|A) = 0.01

\implies P(B|A) = 0.99

\implies P(B^\mathsf{c}|A^\mathsf{c}) = 0.95

P(B|A^\mathsf{c}) = 0.05

+

B

\Omega

A

P(A) = 0.1

P(B^\mathsf{c}|A) = 0.01

\implies P(B|A) = 0.99

\implies P(B^\mathsf{c}|A^\mathsf{c}) = 0.95

P(B|A^\mathsf{c}) = 0.05

A

A \cap B

A \cap B^\mathsf{c}

A^\mathsf{c} \cap B

A^\mathsf{c} \cap B^\mathsf{c}

A^\mathsf{c}

B|A

B^\mathsf{c}|A

B|A^\mathsf{c}

B^\mathsf{c}|A^\mathsf{c}

P(A \cap B \cap C) = P((A \cap B) \cap C)

Let~(A \cap B)=X

\therefore P(A \cap B \cap C) = P(X \cap C)

\therefore P(A \cap B \cap C) = P(X)\cdot P(C|X)

\therefore P(A \cap B \cap C) = P(A\cap B)\cdot P(C|A \cap B)

\therefore P(A \cap B \cap C) = P(A)\cdot P(B|A)\cdot P(C|A \cap B)

P(A \cap B \cap C \cap D)

= P(A)\cdot P(B|A)\cdot P(C|A \cap B)\cdot P(D|A \cap B \cap C)

P(A_1 \cap A_2 \cap \dots \cap A_n)

= P(A_1)\prod_{i=2}^{n} P(A_i|A_1\dots A_{i-1})

p = \frac{4 \choose 3}{52 \choose 3} = \frac{\frac{4!}{1!~3!}}{\frac{52!}{49!~3!}} = \frac{4*3*2}{52*51*50}

A_i

P(A_1\cap A_2\cap A_3)

= P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_2 \cap A_1)

P(A_1)

= \frac{4}{52}

P(A_2|A_1)

= \frac{3}{51}

P(A_3|A_1\cap A_2)

= \frac{2}{50}

= \frac{4*3*2}{52*51*50}

A_1

A_5

A_4

A_3

A_2

A_6

A_7

B

A_1, A_2, \cdots A_n

\Omega

\Omega

A_1 \cup A_2, \cup \dots \cup A_n = \Omega

A_i \cap A_j = \phi~\forall i \neq j

B = (B \cap A_1) \cup (B \cap A_2) \cup \dots \cup (B \cap A_n)

P(B) = P(B \cap A_1) + P(B \cap A_2) + \dots + P(B \cap A_n)

P(B) = \sum_{i=1}^{n} P (A_i) \cdot P(B | A_i)

P(B)

= P(A_1)\cdot P(B|A_1) + P(A_2)\cdot P(B|A_2) + \dots + P(A_n)\cdot P(B|A_n)

+

B

\Omega

A

P(A) = 0.1

P(B^\mathsf{c}|A) = 0.01

\implies P(B|A) = 0.99

\implies P(B^\mathsf{c}|A^\mathsf{c}) = 0.95

P(B|A^\mathsf{c}) = 0.05

P(B) = ?

P(B) = P(A)P(B|A) + P(A^\mathsf{c})(B(|A^\mathsf{c})

\therefore P(B) = 0.1*0.99 + 0.9*0.05 =0.144

P(A_1)=P(A_2)=P(A_3) = \frac{1}{3}

P(B|A_1) = 0.3

P(B|A_2) = 0.6

P(B|A_3) = 0.75

P(B^\mathsf{c}) = ?

A_i:

= P(A_1)P(B^\mathsf{c}|A_1) + P(A_2)P(B^\mathsf{c}|A_2) + P(A_3)P(B^\mathsf{c}|A_3)

= \frac{1}{3}\cdot0.7 + \frac{1}{3}\cdot0.4 + \frac{1}{3}\cdot0.25 = 0.45

P(A_1|B) = ?

A_i:

P(A_1|B) = \frac{P(A_1 \cap B)}{P(B)}

P(A_1|B) = \frac{P(A_1 \cap B)}{P(A_1)\cdot P(B|A_1) + P(A_2)\cdot P(B|A_2) + P(A_3)\cdot P(B|A_3)}

P(A_1 \cap B) = P(A_1)P(B|A_1) \\
P(A_1 \cap B) = P(B)P(A_1|B)

P(A_1)=P(A_2)=P(A_3) = \frac{1}{3}

P(B|A_1) = 0.3

P(B|A_2) = 0.6

P(B|A_3) = 0.75

P(A_1)=P(A_2)=P(A_3) = \frac{1}{3}

P(B|A_1) = 0.3

P(B|A_2) = 0.6

P(B|A_3) = 0.75

P(A_1|B) = ?

A_i:

P(A_1|B) = \frac{P(A_1 \cap B)}{P(B)}

P(A_1|B) = \frac{P(A_1)P(B|A_1)}{P(A_1)\cdot P(B|A_1) + P(A_2)\cdot P(B|A_2) + P(A_3)\cdot P(B|A_3)}

P(A_1 \cap B) = P(A_1)P(B|A_1) \\
P(A_1 \cap B) = P(B)P(A_1|B)

= 0.182

P(A_1)=P(A_2)=P(A_3) = \frac{1}{3}

P(B|A_1) = 0.3

P(B|A_2) = 0.6

P(B|A_3) = 0.75

P(A_3|B) = ?

A_i:

P(A_3|B) = \frac{P(A_3 \cap B)}{P(B)}

P(A_3|B) = \frac{P(A_3)P(B|A_3)}{P(A_1)\cdot P(B|A_1) + P(A_2)\cdot P(B|A_2) + P(A_3)\cdot P(B|A_3)}

P(A_3 \cap B) = P(A_3)P(B|A_3) \\
P(A_3 \cap B) = P(B)P(A_3|B)

= 0.45

P(A_1)P(B|A_1) = P(B)P(A_1|B)

P(B) = P(A_1)\cdot P(B|A_1) + P(A_2)\cdot P(B|A_2) + P(A_3)\cdot P(B|A_3)

P(A_1|B) = \frac{P(A_1)\cdot P(B|A_1) }{\sum_{i=1}^nP(A_i)P(B|A_i)}

P(A) = 0.01

P(B|A) = 0.95

P(B|A^\mathsf{c}) = 0.05

P(A|B) = ?

P(A|B) = \frac{P(A)P(B|A)}{P(A)P(B|A) + P(A^\mathsf{c})P(B|A^\mathsf{c})} = 0.18

A

A^\mathsf{c}

10000

9900

100

495

95

A

A^\mathsf{c}

10000

9000

1000

450

990

+

B

\Omega

A

P(A) = 0.1

P(B^\mathsf{c}|A) = 0.01

\implies P(B|A) = 0.99

\implies P(B^\mathsf{c}|A^\mathsf{c}) = 0.95

P(B|A^\mathsf{c}) = 0.05

P(A|B) = ?

P(A|B) = \frac{P(A)P(B|A)}{P(A)P(B|A) + P(A^\mathsf{c})P(B|A^\mathsf{c})} = 0.6875

P(A_i|B) = \frac{P(A_i)\cdot P(B|A_i) }{\sum_{j=1}^{n}P(A_j)P(B|A_j)}

P(A|B) = \frac{P(A \cap B)}{P(B)}

P(A|B) = \frac{P(A)\cdot P(B|A) }{P(B)}

P(A) = \frac{50}{50+70} = \frac{5}{12}

P(B|A) = \frac{35}{50} = \frac{7}{10}

P(A^\mathsf{c}) = \frac{7}{12}

P(B|A^\mathsf{c}) = \frac{49}{70} = \frac{7}{10}

P(A|B) = ?

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

P(B) = P(B|A)P(A) + P(B|A')P(A') = \frac{7}{10}

= \frac{\frac{7}{10}\frac{5}{12}}{\frac{7}{10}} = \frac{5}{12}

P(A|B) = P(B)

P(A \cap B) = P(B)\cdot P(A|B)

P(A\cap B) = P(A)\cdot P(B)

P(A) = \frac{4}{8}

H | H | H | * | ||

H | H | T | * | * | * |

H | T | H | * | * | * |

H | T | T | * | ||

T | H | H | * | ||

T | H | T | |||

T | T | H | |||

T | T | T |

A

B

A \cap B

P(B) = \frac{3}{8}

P(A \cap B) = \frac{2}{8}

P(A \cap B) \neq P(A)P(B)

A = \{ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \}

B = \{(1,2), (1,4), (1,6), (2,2), (2,4), (2,6),(3,2), (3,4), (3,6), \newline(4,2), (4,4), (4,6), (5,2), (5,4), (5,6), (6,2), (6,4), (6,6)\}

A \cap B = \{(1,6), (3,4), (5,2) \}

P(A) = \frac{6}{36} = \frac{1}{6}

P(A \cap B) = \frac{3}{36}= \frac{1}{12}

P(B) = \frac{18}{36} = \frac{1}{2}

\therefore P(A \cap B) = P(A)\cdot P(B)

P(A) = \frac{1}{4}

P(B) = \frac{1}{3}

P(A \cap B) = P(A)\cdot P(B) = \frac{1}{12}

A_1, A_2, A_3, \dots, A_n

P(A_i \cap A_j) = P(A_i)\cdot P(A_j)~\forall~i\neq j

P(\cap_{i \in I} A_i) = \prod_{i=1}^{n}P(A_i)

A_1, A_2, A_3, \dots, A_n

I \subset \{1,2,3,\dots,n\}

\{1,2,3\}

n = 3

\{1,2\}, \{1, 3\}, \{2,3\}, \{1, 2, 3\}

P(A_1 \cap A_2 ) = P(A_1)\cdot P(A_2)

P(A_1 \cap A_3 ) = P(A_1)\cdot P(A_3)

P(A_2 \cap A_3 ) = P(A_2)\cdot P(A_3)

P(A_1 \cap A_2 \cap A_3 )

= P(A_1)\cdot P(A_2)\cdot P(A_3)

A \perp \!\!\! \perp B

A \not\!\perp\!\!\!\perp B~|~C

P(A\cap B) = P(A)\cdot P(B)

P(A\cap B|C) = P(A|C)\cdot P(B|C)

Compute~P(B|A)~using~P(A\cap B)~and~P(A)

Compute~P(A\cap B)~using~P(A)~and~P(B|A)

Compute~P(B)~using~P(A_1), P(A_2), \dots, P(A_n)

and~P(B|A_1), P(B|A_2), \dots, P(B|A_n)

Compute~P(A|B)~using

P(\cap_{i \in I} A_i) = \prod_{i=1}^{n}P(A_i)