Gate applied Mock Test 3

Question

There are two identical boxes in front of you. In box X, there are 3 white balls and 7 black balls. In box Y there are 7 white balls and 3 black balls. You randomly reach one of the boxes and draw three balls from it one by one. If every time you draw a ball you find it is white, at which of the following stages is the probability the highest that you chose box X?

I. You drew one ball, and it was white.

II. You drew two balls, and they were both white.

III. You drew three balls, and they were all white.

Comments

in 1. When You drew one ball, and it was white.

You can do by finding all 3 probabilites $P(X|W), P(X|W \cap W), P(X|W \cap W\cap W)$ which will take some time, or you can observe and say it's option 1, bcoz ratio of white to total balls is less in X compared to Y, so for drawing white ball everytime is difficult in X compared to Y, hence the chance that that box is X will keep on decreasing.

btw, is option 1 correct ?