Question

Suppose we have two bags with the numbers. Each bag has a total of 100 numbers. In the first bag there are 31 lucky numbers, in the second bag there are 18 lucky numbers. We want to add one more bag with 100 numbers to decrease the probability that a randomly selected number from a random bag is the lucky number. How many lucky numbers should be in the third bag?

Answer 1

There should be at most 24 lucky numbers in the third bag.

Explanation:

Initially, there are 200 numbers. Two bags with 100 each. There are 31+18 = 49 lucky numbers. So there is a 49/200 = 0.245 probability that a randomly selected number from a random bag is the lucky number.

Now with 300 numbers, we want this probability to be lower than 24.5%. So we should solve the following rule of three:

200 - 49

300 - x

`200x = 300*49`

`x = 1.5*49`

`x = 73.5`

With the third bag, the probability will be the same if 73.5-49 = 24.5 lucky numbers are added. So there should be at most 24 lucky numbers in the third bag.