Lily has only red and blue balls in a jar. If she removes three balls

Lily has only red and blue balls in a jar. If she removes three balls, is the probability of getting all red balls greater than the probability of getting at least one blue ball?

We require to know the number of red and blue balls. Also the two probabilities P(all red) and P(at least one blue) will add up to 1.
So knowing anyone of the P to be greater than 1/2 will give a definite answer.


(1) The number of red balls is more than three times the number of blue balls.
A wide range - In 1000 balls, red could be 751 to 999.
Red balls in very high proportion will give YES:
Say 1000 balls out of which 999 are red and one blue - P(all 3 red) = 999C3/1000C3 = 997/1000 and P(at least one Blue) = 999C2*1C1/1000C3 = 3/1000
Exactly 3 times will give a NO:
Say 5 balls out of which 4 are red - P(all 3 red) = 4C3/5C3 = 2/5 and P(at least one Blue) = 4C2*1C1/5C3 = 3/5
Insufficient

(2) Less than (1)/(4) th of the balls in the jar are blue.
This again leaves a wide range in front of us and blue can have varying values. For example out of 1000 balls, blue could vary from 1 to 249.
Red balls in very high proportion will give YES:
Say 1000 balls out of which 999 are red and one blue - P(all 3 red) = 999C3/1000C3 = 997/1000 and P(at least one Blue) = 999C2*1C1/1000C3 = 3/1000
Red balls in not a very high proportion will give a NO:
Say 5 balls out of which 4 are red - P(all 3 red) = 4C3/5C3 = 2/5 and P(at least one Blue) = 4C2*1C1/5C3 = 3/5
Insufficient

Combined
Nothing new



E