Probability that two independent variables with mass function $\mathbf{P}(X = n) = 2^{-n}$ are $> 3$. (Subject GRE Exam 0568 Q.42)

Question

The question and its answer is given in the following picture:

I do not understand the second equality in the solution from where it comes, could anyone explain this for me please?

Answer 1

Independence of $X$ and $Y$ means $$P(X\text{ does thing }1\text{ and } Y \text{ does thing 2})=P(X\text{ does thing 1})P(Y\text{ does thing 2})$$ for all pairs of things.

$X$ and $Y$ have the same distribution means $$P(X\text{ does thing T})=P(Y\text{ does thing T})$$ for all things $T$.

Answer 2

You want the probability X or Y is greater than 3. The other possibility is that X and Y are both less than or equal to 3. So those two possibilities sum to 1. It's a lot easier to calculate the probability X and Y are both less than or equal to 3 (which is 49/64), so do that and subtract it from 1. The reason there are two inequalities is that both conditions must be true.

Answer 3

$X$ and $Y$ are independent therefore $\mathbb P(X \leq 3 \textrm{ and } Y \leq 3) = \mathbb P(X \leq 3) \cdot \mathbb P(Y \leq 3) = \mathbb P(X \leq 3) \cdot \mathbb P(X \leq 3)$ because $X$ and $Y$ have the same probability distribution.

Answer 4

We assume $X$ and $Y$ are independent random variables, and so we can say that $P(X \leq x \ \text{and} \ Y \leq y) = P(X \leq x)P(Y \leq y).$ This is very powerful when used correctly.