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=800−x+JR=800−x+x2600+x=800(600+x)−x(600+x)+x2600+x=4800+800x−600x−x2+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
12 months ago

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Comments

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

A Former Brilliant Member - 12 months ago
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