How do I calculate overall probability of an occurrence with unique probabilities at different attempts?

Question

cenario: You have marbles of 2 colors, red and blue. You put one red and one blue marble in a bag and take one at random. The probability of getting a red marble is obviously 50%. But then you add 5 more blue marbles in the bag and take one at random again. The probability of getting red for that attempt alone is obviously 1/7, but how do I calculate the overall probability of getting a red marble at least once through both attempts whereas each attempt has a different probability?

Attempt 1: 2 marbles, 1 red, 1 blue

Attempt 2: 7 marbles, 2 red, 5 blue

Attempt 3: 10 marbles, 3 red, 7 blue

It doesn't take a mathematician to figure out the probabilities for each individual attempt, but in the given scenario, how is the overall probability calculated?

By the way, I already figured it out using pictures: 105/140 chance. But is there some formula I can use for such math problems like above?

Answer 1

Just note that $$1-p = (1-p_1)(1-p_2)(1-p_3)$$

This simply says that for the experiment to fail, you must fail on each attempt.

So the probability of success on the combined experiment is

$$p = 1-(1-p_1)(1-p_2)(1-p_3).$$

(Here, $p_j$ is the probability of success (drawing a red marble) on the $j^{\textrm{th}}$ attempt. I also assume you are drawing with replacement.)

Answer 2

Here you can break it into cases:

$$(\mbox{happens on first attempt})+(\mbox{doesn't happen on first attempt} )\cdot(\mbox{happens on second attempt} ) +(\mbox{doesn't happen on first or second attempt})\cdot (\mbox{happens on third attempt}) $$