On Daily challenge problem

I answered this Daily challenge problem correct but was marked as incorrect. I will give a proof for my answer below

Assumptions:

  • Capacity of candies of the cup is constant.

  • Candies are mixed well, i.e. the concentration of candies is uniform all over the jar.

  • Candies which are more concentrated are more probable to be selected if you pick any candy randomly from a mixture.

Let the capacity of the cup be of xx candies

After first step:

Total candies in Jar BB=600+x600+x

Number of Red candies = xx

Number of Green candies = 600600

Concentration of Red candies = CAR=x600+xC_{AR}=\frac{x}{600+x}

Concentration of Green candies = CAG=600600+xC_{AG}=\frac{600}{600+x}

Taking one cup of candies from jar BB:

Number of Green candies in the cup = JG=CAG×x=600x600+xJ_G=C_{AG} \times x=\frac{600x}{600+x}

Number of Red candies in the cup = JR=CAR×x=x2600+xJ_R=C_{AR} \times x=\frac{x^2}{600+x}

After second step:

Total number of candies in jar AA=800800

Number of Red candies=NR=800x+JR=800x+x2600+x=800(600+x)x(600+x)+x2600+x=4800+800x600xx2+x2600+x=4800+200x600+xN_R=800-x+J_R=800-x+\frac{x^2}{600+x}=\frac{800(600+x)-x(600+x)+x^2}{600+x}=\frac{4800+800x-600x\cancel{-x^2+x^2}}{600+x}=\frac{4800+200x}{600+x}

Number of Green candies = NG=JG=600x600+xN_G=J_G=\frac{600x}{600+x}

Concentration of Red candies = CBR=NR800=4800+200x800(600+x)C_{BR}=\frac{N_R}{800}=\frac{4800+200x}{800(600+x)}

Concentration of Green candies=CBG=NG800=600x800(600+x)C_{BG}=\frac{N_G}{800}=\frac{600x}{800(600+x)}

After third step:

Concentration of Red candies in the mixture = CAR+CBR2=12(x600+x+4800+200x800(600+x))=12(800x+4800+200x600(800+x))=2400+500x600(800+x)\frac{C_{AR}+C_{BR}}{2}=\frac{1}{2}(\frac{x}{600+x}+\frac{4800+200x}{800(600+x)})=\frac{1}{2}(\frac{800x+4800+200x}{600(800+x)})=\boxed{\frac{2400+500x}{600(800+x)}}

Concentration of Green candies in the mixture=CAG+CBG2=12(600600+x+600x800(600+x))=12(4800+600x600(800+x))=2400+300x600(800+x)\frac{C_{AG}+C_{BG}}{2}=\frac{1}{2}(\frac{600}{600+x}+\frac{600x}{800(600+x)})=\frac{1}{2}(\frac{4800+600x}{600(800+x)})=\boxed{\frac{2400+300x}{600(800+x)}}

If you compare the concentrations of both the candies in the last mixture, you can conclude that Red candies are more concentrated than Green candies

\therefore Red candies are more likely to be selected in the fourth step.

Note by Zakir Husain
3 months, 2 weeks ago

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Great! @Zakir Husain. I agree with you but shouldn't you report this as a bug?

A Former Brilliant Member - 3 months, 2 weeks ago

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