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A sample space consists of 9 elementary ...

A sample space consists of 9 elementary event `E_1, E_2, E_3 ..... E_8, E_9` whose probabilities are `P(E_1) = P(E_2) = 0. 08` ,`P(E_3) = P(E_4)=P(E_5) = 0. 1`, `P(E_6) = P(E_7) = 0. 2` ,`P(E_8) = P(E_9) = 0. 07`. Suppose `A = {E_1,E_5,E_8}`, `B = {E_2, E_5, E_8, E_9}`. Compute `P(A)`, `P(B)` and `P(AnnB)`. Using the addition law of probability, find `P(AuuB)`. List the composition of the event `AuuB`, and calculate, `P(AuuB)` by adding the probabilities of the elementary events. Calculate `P(barB)` from `P(B)`, also calculate `P(barB)` directly from the elementary events of `barB`.

Text Solution

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(a) `P(A) = P(E_(1)) + P(E_(5)) + P(E_(8))`
`=0.09 + 0.1 + 0.06 = 0.25`
(b) `P(B) = P(E_(2)) + P(E_(5)) + P(E_(8)) = P(E_(9))`
`= 0.09 + 0.1 + 0.06 + 0.06 = 0.31`
`P(A uu B) = P(A) + P(B) - P(A nn B)`
Now, `A nn B = {E_(5), E_(8)}`
`therefore P(A nn B) = P(E_(5)) + P(E_(8)) = 0.1 + 0.06 = 0.16`
`therefore P(A uu B) = 0.25 + 0.31 - 0.16 = 0.40`
(c ) `A uu B = {E_(1), E_(2), E_(5), E_(8), E_(9)}`
`P(A uu B) = P(E_(1)) + P(E_(2)) + P(E_(5)) + P(E_(8)) + P(E_(9))`
`=0.09 + 0.09 + 0.1 +0.06 + 0.06 = 0.40`
(d) `because P(bar(B)) = 1- P(B) = 1 - 0.31 = 0.69`
and `bar(B) = {E_(1), E_(3), E_(4), E_(6), E_(7)}`
`therefore P(bar(B)) = P(E_(1)) + P(E_(3)) + P(E_(4)) + P(E_(6)) + P(E_(7))`
= 0.09 + 0.1 + 0.1 + 0.2 + 0.2 = 0.69
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