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An urn contains r red balls and b black ...

An urn contains r red balls and b black balls. Now, match the following lists:

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The correct Answer is:
`a to q , r ; b to r ; c to p ; d to p , q, r, s`
`a to q,r, b to r,c to p,d toa,b,c,d.`
a. `(""^(r)C_(2))/(""^(r+b)C_(2))=1/2(r-1)=(r-b)(r+b-1)`
`=2r(r-1)=(r+b)(r+b-1)`
`or 2r^(2)-2r=r^(2)+(2b-1)r+b^(2)-1`
`or r^(2)-(1+2b)+1-b^(2)=0`
`or b^(2)+2br+r-r^(2)-1=0`
`or b=(-2r+-sqrt(4r^(2)-4(r-r^(2)-1)))/(2)`
`=-r+-sqrt(2r^(2)-r+1)`
Since b is integer, possible values of r are 3 and 8.
b. `""^(4)C_(2)((r)/(r+10))((10)/(r+10))^(2)=3/8`
c. `((r)/(r+10))^(2)((10)/(r+10))=1/16impliesr=10`
d. Probability of getting n red balls in 2n draws in always equal to probability of getting exactly n black blalls in 2 n draws for any value of r and b, hence the ratio r/b can be `10,3,8,2.`
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