A commitee is formed by randomly selecting from eight nurses and two doctors. What is the minimum commitee size that ensures at least a 90% probablity that it will not be comprised of nurses only?

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Let committee is formed by selecting only nurse and size of the committee be x.

E= allmembers in committee are nurse.

`barE=` all members in committee are not nurse

`P(E)=(^8C_n)/(^10C_n)`

`=((10-x)(9-x))/90`

`Thus`

`P(barE)=1-P(E)=1-((10-x)(9-x))/90`

`=(19x-x^2)/90`

By given condition,

`P(barE)>=.9`

`19x-x^2>=.9xx90`

`x^2-19x+81<=0`

`(x-(19+sqrt(37))/2)(x-(19-sqrt(37))/2)<=0`

`=>`

Either  `x<=(19+sqrt(37))/2 and x>=(19-sqrt(37))/2`

`=> 6.45(approx)<=x<=12.54 (approx)`

`or`

`x>=(19+sqrt(37))/2 and x<=(19-sqrt(37))/2`

`` which  has no solution.

Thus size of committee may either 7 or 8 or 9 or 10.

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