The urn contains a ball of unknown color with equal probability: white or red. A white and a red ball is dropped

The urn contains a ball of unknown color with equal probability: white or red. A white and a red ball is dropped into the urn, and after mixing, one ball is drawn at random. Find the probability that balls of the same color remain in the urn?

Let be:

Н1 – there was a white ball in the urn;

P (H1) = 1/2 = 0.5;

H2 – there was a red ball in the urn;

P (H1) = 1/2 = 0.5.

Random event A – a ball was pulled out at random after a red and white ball was placed in the urn.

Let’s apply the formula of total probability:
P (A) = P (H1) * P (A | H1) + P (H2) * P (A | H2) = 1/2 * 1/2 + 1/2 * 1/2 = 1/4 + 1 / 4 = 1/2;
Let’s apply Bayes’ formula.
P (H1 | A) = P (H1) * P (A | H1) / P (A) = 2/3.