Question

A sample space consists of 9 elementary events E₁, E₂, E₃, ..., E₉ whose probabilities are
P(E₁) = P(E₂) = 0.08, P(E₃) = P(E₄) = P(E₅) = 0.1, P(E₆) = P(E₇) = 0.2, P(E₈) = P(E₉) = 0.07
Suppose A = {E₁, E₅, E₈}, B = {E₂, E₅, E₈, E₉}   

 Calculate \[P\left( \bar{ B} \right)\]  from P(B), also calculate \[P\left( \bar{ B } \right)\]  directly from the elementary events of \[\bar{ B } \] .

 

Answer 1

Let S be the sample space of the elementary events.
S = {E₁, E₂, E₃, ..., E₉}
Given:
A = {E₁, E₅, E₈}
B = {E₂, E₅, E₈, E₉}
P(E₁) = P(E₂) = 0.08, P(E₃) = P(E₄) = P(E₅) = 0.1, P(E₆) = P(E₇) = 0.2, P(E₈) = P(E₉) = 0.07 

 \[P\left( B \right) = 1 - P\left( B \right) = 1 - 0 . 32 = 0 . 68\]  [From (i)]
Also, we know that  \[\bar{ B } \]= S − B = {E₁, E₃, E₄, E₆, E₇}

∴ \[P\left( \bar{B} \right)\] = P(E₁) + P(E₃) + P(E₄) + P(E₆) + P(E₇)

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