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 The Multiplication rule for two events says:
 P(AB) = P(A) P(B  A)
 The Multiplication rule extends to 3 Events:
 P(ABC) = P(AB)P(C  AB) = P(A) P(B  A) P(C  AB)

3

 Similarly, it extends to n events:
 P(A_{1} A_{2}
A_{n}) = P(A_{1}
A_{n1})P(A_{n}A_{1}
A_{n1})
 = P(A_{1}) P(A_{2}A_{1}) P(A_{3}A_{1}
A_{2})
P(A_{n} A_{1}
A_{n1})

4

 We roll two dice. What is the chance that we will roll out Shesh Besh: for the first time on the
nth roll?

5

 This is called a Geometric Distribution
 with parameter p=1/36.

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7

 If there are n students in the class, what is the chance that at least
two of them have the same birthday?
 P(at least 2 have same birthday) =
 1 P(No coinciding birthdays).
 Let B_{i} be the birthday of student number i.
 The probability of no coinciding birthdays is:
 P(B_{2} Ο {B_{1}}
& B_{3} Ο {B_{1},B_{2}} &
& B_{n}
Ο {B_{1},
,B_{n1}}).

8

 P(B_{2} Ο {B_{1}} & B_{3} Ο {B_{1},B_{2}}
&
& B_{n} Ο {B_{1},
,B_{n1}}).

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 P(at least 2 have same birthday) =
 1 P(No coinciding birthdays) =

10

 log(P(No coinciding birthdays))=

11

 P(No coinciding birthdays)

12

 Probability of no coinciding birthday as a function of n

13

 Recall that A and B are independent if:
 P(BA)=P(BA^{c}) = P(B);
 We say that A,B and C are independent if:
 P(CAB)= P(CA^{c}B) = P(CA^{c}B^{c}) = P(CAB^{c})
=P(C)

14

 The events A_{1},
,A_{n} are independent if
 P(A_{i}  B_{1},
B_{i1},B_{i+1},
B_{n})=
P(A_{i})
 for B_{i} = A_{i} or A_{i}^{c}
 This is equivalent to following multiplication rules:
 P(B_{1} B_{2}
B_{n}) = P(B_{1}) P(B_{2})
P(B_{n})
 for B_{i} = A_{i} or A_{i}^{c}

15

 Question: Consider the events A_{1},
,A_{n}.
 Suppose that for all i and j the events A_{i} and A_{j}
are independent.
 Does that mean that A_{1},
,A_{n} are all independent?

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 I pick one of these people at random. If I tell you that its a girl,
there is an equal chance that she is a blond or a brunet; she has blue
or brown eyes. Similarly for a
boy.
