Draw me until you match me

Assume that the number which came out the first comes out again on the second turn. The probability is $\frac{1}{4}$ .

There are also cases when the number which came out the first comes out again on the third turn. (On the second turn, other numbers should come out.) The probability is $\frac{3}{4} \times \frac{1}{4} = \frac{3}{16}$

fourth turn $\frac{3}{4} \times \frac{2}{4} \times \frac{1}{4} = \frac{3}{32}$ (each number, excluding the number which came out the first, should not come out again.)

fifth turn $\frac{3}{4} \times \frac{2}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{3}{128}$

So the total probability is $\frac{1}{4}+\frac{3}{16}+\frac{3}{32}+\frac{3}{128}=\frac{71}{128}$ . Hence the answer is 71+128= 199 .

By the pigeon-hole principle, if you draw 5 balls, at least 2 will be of the same colour. And hence we have 4 cases:

Case (i): The first ball you draw is same as the second ball. Trivially, the probability for this case is $1/4$

Case (ii): The first ball you draw is the same as the third ball. That means the second ball you draw must be a different colour from the first and last draw. Hence probability for this case is:

probability that second draw is diff (hence 3 colours to choose from) * probability that third draw same as first draw

= $3/4 \times 1/4$ = 3/16

Case (iii): The first ball you draw is the same as your fourth ball. Same analysis as above shows that the second ball needs to be different from the first and fourth (and hence 3 choices) and your third ball needs to be different from the first and fourth along with the second (and hence $3 - 1 = 2$ choices) . Whence,

$3/4 \times 2/4 \times 1/4 = 3/32$

Case (iv): Well, this time it is seriously the same scenario as case (iii). The only additional change for this case is that the fourth ball is being limited to $2 - 1 = 1$ colours. Whence,

$3/4 \times 2/4 \times 1/4 \times 1/4 = 3/128$

In summary, the total probability is:

case (i) + case (ii) + case (iii) + case (iv) = $1/4 + 3/16 + 3/32 + 3/128 = 71/128$

Therefore, $a + b = 71 + 128 = 199$