Instead of thinking about probability, think in terms of how many different pairs are possible. Since each pair is equally likely, counting the pairs and comparing them is all we need to do.

There are 20 balls altogether, so there are 20 options for the first ball. There are 19 options for the second ball, but since a pair is the same pair if they are drawn in reverse order, we can halve this product. So the total number of possible pairs is 20 x 19 / 2 = 190.

We can perform the exact same calculations for each set of same-coloured balls within the bag. For instance if there were 15 red balls, then there are 15 x 14 / 2 = 105 pairs that would be made up of two red balls. This is already more than half of 190, so there cannot be 15 balls of a particular colour.

Instead let’s try 14 red balls. This gives us 14 x 13 / 2 = 91 red-red pairs. This is close to the 190 / 2 = 95 pairs we need, so we are on the right track.

If there were 3 balls of a second colour (say, blue) then there would be 3 x 2 / 2 = 3 possible blue-blue pairs. Add these to the 91 red-red pairs and we are almost there.

Add in 2 balls of a third colour (say, white) then there is one white-white pair. This makes up all of the 95 pairs we needed, but we have only used 19 balls, so we need a single ball of a fourth colour (say, green) to make it up to 20.

14 RED, 3 BLUE, 2 WHITE, 1 GREEN.

If you play about with other possibilities you will find that this arrangement, (14,3,2,1), is the only way of exactly achieving the aim of exactly 50% chance of a matching pair.