Recitation 21 Problem 2: Expected value

Question

I came from an engineering background and I am currently self-learning MIT 6.042 Discrete Mathematics.

I am having difficulty in understanding Recitation 21 Problem 2c's solution. In particular, I do not understand how did the joint event of two and three events being reduced to just one as highlighted in green and blue in the snapshot below. Appreciate if someone could kindly explain the reasoning behind the simplification or point me to any relevant materials that I could read on.

The question and solution provided to question 2c are as follow:

Problem 2

Solution to Problem 2c

Thanks in advance.

Answer 1

What is the probability you roll only one time (i.e. no doubles) What is the expectation in this scenario?

What is the probability you roll doubles on the first roll and no doubles on the second roll? Two rolls and what is the expectation for each?

What is the probability you roll doubles on the first two rolls and no doubles on the third roll?
Expectation?

And finally what is the probability you go to jail?

Do all of your probabilities sum to 1? If not, then you have missed something.

Sum the expectation for each scenario weighted by the probability of each scenario.