Introduction to Probability and Statistics for Engineers and Scientists

80 Chapter 3:Elements of Probability

P(AiAj)= 1 −P(Aci∪Acj)

= 1 −[P(Aci)+P(Acj)−P(AciAcj)]

= 1 −(1−pi)k−(1−pj)k+P{no coupon is typeior typej}

= 1 −(1−pi)k−(1−pj)k+(1−pi−pj)k

where the final equality follows because each of thekcoupons is, independently, neither
of typeior of typejwith probability 1−pi−pj. Consequently,

P(Aj|Ai)=

1 −(1−pi)k−(1−pj)k+(1−pi−pj)k

1 −(1−pi)k

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Problems..........................................................

1. A box contains three marbles — one red, one green, and one blue. Consider an
   experiment that consists of taking one marble from the box, then replacing it in
   the box and drawing a second marble from the box. Describe the sample space.
   Repeat for the case in which the second marble is drawn without first replacing
   the first marble.


2. An experiment consists of tossing a coin three times. What is the sample space
   of this experiment? Which event corresponds to the experiment resulting in more
   heads than tails?


3. LetS={1, 2, 3, 4, 5, 6, 7},E={1, 3, 5, 7},F={7, 4, 6},G={1, 4}. Find
   (a)EF; (c)EGc; (e)Ec(F∪G);
   (b)E∪FG; (d)EFc∪G; (f )EG∪FG.


4. Two dice are thrown. LetEbe the event that the sum of the dice is odd, letFbe
   the event that the first die lands on 1, and letGbe the event that the sum is 5.
   Describe the eventsEF,E∪F,FG,EFc,EFG.


5. A system is composed of four components, each of which is either working or failed.
   Consider an experiment that consists of observing the status of each component,
   and let the outcome of the experiment be given by the vector (x 1 ,x 2 ,x 3 ,x 4 ) where
   xiis equal to 1 if componentiis working and is equal to 0 if componentiis failed.
   (a) How many outcomes are in the sample space of this experiment?
   (b) Suppose that the system will work if components 1 and 2 are both working,
   or if components 3 and 4 are both working. Specify all the outcomes in the
   event that the system works.
   (c) LetEbe the event that components 1 and 3 are both failed. How many
   outcomes are contained in eventE?