I\'ve been working on this question for four days. Please bekind to help. Urn 1

Question

I've been working on this question for four days. Please bekind to help. Urn 1 contains 7 white and 3 black balls, whereas Urn 2contains 4 white and 6 black balls. From each urn a ball isselected at random, independently, and put in the other urn. Twoballs are then selected at random from Urn 1. Let X denote thenumber of black balls among these two. Find E(X) and V(X). Thank you for your time! I've been working on this question for four days. Please bekind to help. Urn 1 contains 7 white and 3 black balls, whereas Urn 2contains 4 white and 6 black balls. From each urn a ball isselected at random, independently, and put in the other urn. Twoballs are then selected at random from Urn 1. Let X denote thenumber of black balls among these two. Find E(X) and V(X). Thank you for your time!

Explanation / Answer

First lets simplify the question Both urns have 10balls in all where you have to interchange 2talls urn 1 has 7W3B balls and urn 2 has 4W6B balls if you interchange two balls randomly andindependently you will have two urns with 10 balls each again butdifferent colours depending upon the balls interchanged now you will have 3 possibilites for a new urn(not 4) [ we will ony consider urn 1 since we need it] now consider various interchanges to get 3 possibilities fornew urn 1 suppose you pick black ball from urn 1 and put it in urn 2 andin turn you pick a white ball from urn 2 and put it in urn 1 this way a white ball increases in the new urn 1 and now youwill have 8W2B balls in urn 1 vice versa interchange will lead to 6W4B balls a white ballwill reduce in number finally if you pick both balls of same colour and exchangethen you will still have same colours combo i.e. 7W3B we name these events to be A, B, C respectively nowP(A)=3/10*4/10=12/100
P(B)=7/10*6/10=42/100 P(C)=7*4/100 +3*6/100= 46 eitherboth white balls are interchanged or both black balls areinterchanged suppose you select new urn A then we have to find P(X/A) where X=0,1,2 out of two balls drawn none are balck colour exactly 1 isblack or both are black now urn A has 8W and 2B X/A             0                                     1                          2 P(X/A)      C(8,2)/C(10,2)           8*2/C(10,2)          1/C(10,2)                              28/45                16/45                     1/45 E(X/A) =0 +16/45+ 2/45 =18/45 similarly for urn B with 6W4B P(X=0/B) = 15/45, P(X=1/B)=24/45 P(X=2/B) 6/45 E(X/B)= 0+ 24/45 +12/45= 36/45 and for urn C with 7W and 3B balls P(X=1/C)= 21/45   and P(X=2/C)= 3/45 E(X/C)= 21/45 +6/45= 27/45 now we need to find out E(X) we know E(X)=E(E(X/Y)) in this case it is E(X)= E(E(X/A)) + E(E(X/B)) + E(E(X/C)) E(X)= P(A)*E(X/A)+  P(B)*E(X/B)+ P(C)*E(X/C)          =12/100*18/45+ 42/100*36/45 +46*27/45

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