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Question

A sample space consists of 9 elementary events

E1,E2E3,....,E8,E9

whose probabilities are

P(E1)=P(E2)=0.08,

P(E3)=P(E4)=0.1,

P(E6)=P(E7)=0.2

P(E8)=P(E9)=0.07

Suppose  A={E1,E5,E8},

B ={E2,E5,E8,E9}

(i) Compute P(A),P(B), P(AB)

(ii) Using the addition law of probability, find P(AB)

(iii) List the composition of the even AB, and calculate

P(AB) by adding the probabilities of the elementary events.

(iv) Calculate P(¯¯¯¯B) from P(B), also calculate

P(¯¯¯¯B) directly from the elementary events of ¯¯¯¯B


Solution

Given,S =  {E1,E2,E3E4,E5,E6,E7,E8,E9}

A = {E1,E5,E8}, B = {E2,E5,E8,E9}

P(E1)=P(E2)=0.08,

P(E3)=P(E4)=0.1,

P(E6)=P(E7)=0.2

P(E8)=P(E9)=0.07

(i) P(A)=P(E1)+P(E5)+P(E8)

=0.08+0.1+0.07=0.25

(ii) P(AB)=P(A)+P(B)P(AB)                                 ....(i)

Now, P(B)=P(E2)+P(E5)+P(E8)+P(E9)

=0.08+0.1+0.07+0.07=0.32

AB={E5,E8}

P(AB)=P(E5)+P(E8)=0.1+0.7=0.17

 On substituting these values in Eq. (i), we get

P(AB)=0.25+0.32-0.17=0.40

(iii) AB={E1,E2,E5,E8,E9}

P(AB)=P(E1)+P(E2)+P(E5)+P(E8)+P(E9)=0.08+0.08+0.1+0.07+0.07=0.40

(iv) P(¯¯¯¯B) =1P(B)10.32=0.68

¯¯¯¯B={E1,E3,E4,E6,E7}

P(¯¯¯¯B)=P(E1)+P(E3)+P(E4)+P(E6)+P(E7)

=0.08+0.1+0.1+0.2+0.2=0.68

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