Question 1199439
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A box A contains 4 gold rings and 3 diamond rings. 
A box B contains 2 gold rings and 5 diamond rings. 
A ring is picked at random from box A and placed in box B. 
A ring is then picked at random from box B and placed in box A. 
Determine the probability that:
(a) box A has the same number of gold and diamond rings as it did initially. 
(b) box A has more gold rings than it did initially.
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(a)  Case (a) may happen if and only if any one of these two two-step events will happen

          (1) EITHER a gold    ring is picked from A to B and a gold    ring is then picked from B to A,

          (2)   OR   a diamond ring is picked from A to B and a diamond ring is then picked from B to A.


         The probability of the two-step event (1) is P = {{{(4/7)*(3/8)}}} = {{{12/56}}}.

         The probability of the two-step event (2) is P = {{{(3/7)*(6/8)}}} = {{{18/56}}}.


         Thus the probability of case (a) is  P(a) = {{{12/56 + 18/56}}} = {{{30/56}}} = {{{15/28}}}.   <U>ANSWER</U>



(b)  Case (b) may happen if and only if any this one two-step event will happen

          a diamond ring is picked from A to B and a gold ring is then picked from B to A.


         The probability of this two-step event is P(b) = {{{(3/7)*(2/8)}}} = {{{3/28}}}.    <U>ANSWER</U>.
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Solved and fully explained.