Multiplication rule for two events

The following are the cases possible when the dies are thrown and the sum of the numbers on the two faces is $$4$$:

1 and 3
First die 1 and the second die 3 = $$\dfrac{2}{6}\times \dfrac{2}{6} = \dfrac{4}{26}$$
First die 3 and the second die 1 = $$\dfrac{1}{6}\times \dfrac{1}{6} = \dfrac{1}{36}$$

2 and 2
First die 2 and the second die 2 = $$\dfrac{2}{6}\times \dfrac{2}{6} = \dfrac{4}{36}$$

The probability of getting a sum of $$4$$ in all: $$\dfrac{4+1+4}{36} = \dfrac{1}{4}$$

The following are the cases possible when the dies are thrown and the sum of the numbers on the two faces is $$5$$:

1 and 4
First die 1 and the second die 4: $$\dfrac{2}{6}\times \dfrac{1}{6} = \dfrac{2}{36}$$
First die 4 and the second die 1: $$\dfrac{1}{6}\times \dfrac{1}{6} = \dfrac{1}{36}$$

2 and 3
First die 2 and the second die 3: $$\dfrac{2}{6}\times \dfrac{2}{6} = \dfrac{4}{36}$$
First die 3 and the second die 2: $$\dfrac{1}{6}\times \dfrac{2}{6} = \dfrac{2}{36}$$

The probability of getting a sum of $$5$$ in all: $$\dfrac{2+1+4+2}{36} = \dfrac{1}{4}$$

Thus, the combined probability of getting a sum of either $$4$$ or $$5$$ when the two dice are thrown is $$\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{1}{2}$$