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  • A sample space consists of 9 elementary outcomes e1 e up to e9 whose probabilities are probability of event e1 equal to probability of event e2 equal to 08 probability of event e3 equal to probability of event e4 equal to probability of event e5

A sample space consists of 9 elementary outcomes e1, e2, ..., e9 whose probabilities are

P(e1) = P(e2) = .08, P(e3) = P(e4) = P(e5) = .1
P(e6) = P(e7) = .2, P(e8) = P(e9) = .07
Suppose A = {e1, e5, e8}, B = {e2, e5, e8, e9}
(a) Calculate P (A), P (B), and P (A ∩ B)
(b) Using the addition law of probability, calculate P (A ∪ B)
(c) List the composition of the event A ∪ B, and calculate P (A ∪ B) by adding the probabilities of the elementary outcomes.
(d) Calculate P (\bar{B}) from P (B), also calculate P (\bar{B}) directly from the elementary outcomes of \bar{B}

Answers (1)

Given data:

S = {e1, e2, e3, e4, e5, e6, e7, e8, e9}

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

  1.  P (A), P (B) & P (A ∩ B)

A = {e1, e5, e8}  …….. (given)

Thus,

P (A) = P (e1) + P (e5) + P (e8)

         = 0.08 + 0.1 + 0.07

                  = 0.25

Now, B = { e2, e5 e8, e9}    ……. (given)

P (B) = P (e2) + P (e5) + P (e8) + P (e9)

Now, P (A ∩ B)

A ∩ B = {e5, e8}

Thus, P (A ∩ B) = P (e5) + P (e8)

                        = 0.1 + 0.07

                        = 0.17

  1. P (A U B)

P (A) = 0.25

P (B) = 0.32

P (A ∩ B) = 0.17

Now, P (A U B) = P (A) + P (B) – P (A ∩ B)        ……. (by general addition rule)

= 0.25 + 0.32 – 0.17

= 0.40

  (c) A = {e1, e5, e8}

       B = { e2, e5 e8, e9}        ……. (given)

Thus, A U B = {e1, e2, e5, e8, e9}

Thus, P (A U B) = P (e1) + P (e2) + P (e5) + P (e8) + P (e9)

                        = 0.08 + 0.08 + 0.1 + 0.07 + 0.07

                        = 0.40

(d) P (\bar{B})

P (\bar{B}) = 1 – 0.32

            = 0.68

Now, we have,

B = { e2, e5 e8, e9}       

Thus, \bar{B} = { e1, e3, e4, e6, e7}

P (\bar{B}) = P (e1) + P (e3) + P (e4) + P (e6) + P (e7)

            = 0.08 + 0.1 + 0.1 + 0.2 +0.2

            = 0.68

Posted by

infoexpert22

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