Addition Rule of Probability: Example Problems

Getting 2 or 5 in two throws should be $P(2)+P(5)$. $P(2)=1/6,P(5)=1/6$ so the combined so it should be 1/3.
I tried to visualize but not able to do so correctly.
11,12,13,14,15,16, 21,22,23,24,25,26,31,32, ....6,6
total of 36 possibilities.
12,15,21,22,23,24,25,26,31,35,42,45,51,52,53,54,55,56,61,65
out of which 20 possibilities, so the probability should be 20/36 which is not 1/3.
Where am I going wrong?

Okay, so, my teacher gave us this worksheet of "harder/unusual probability questions", and Q.5 is real tough. I'm studying at GCSE level, so it'd be appreciated if all you stellar mathematicians explained it in a way that a 15 year old would understand. Thanks!
So, John has an empty box. He puts some red counters and some blue counters into the box.
The ratio of the number of red counters to blue counters is 1:4
Linda takes out, at random, 2 counters from the box.
The probability that she takes out 2 red counters is 6/155
How many red counters did John put into the box?

I'm studying the Cartesian product, which is bound to the idea of a binary relation. Even with Cartesian products of several sets, n-ary Cartesian products, we have to think combinatorically as two sets at a time, recursively. Is there any sort of operation where there is, say, a triary operation happening. Obviously, when we have $3-<!--\; -\; -->2-<!--\; -\; -->1$ we have to think of associative rules, forcing subtraction to be a binary operator "step-through" affair. There is no $3\oplus <!--\; \oplus \; -->2\oplus <!--\; \oplus \; -->1$ operator that does something in one fell swoop to all three numbers is there? Lisp has (+ 1 2 3) and grade-school math has vertical addition with the plus-sign and a line under the lowest number, but these are not "internally" non-binary. The only thing that takes "three at once" is, yes, a product-based operator, again, an n-ary Cartesian product. Correct? Another example would be playing poker and being dealt five cards. The cards were shuffled, which is a combinatoric permutation of the stack of cards, which are then dealt into hands. Is there anything from probability (or anywhere) that doesn't start with a shuffle permutation therefore binary, rather, just looks at the five cards coming together non-permutation-, non-binary-wise?

I know that there is the addition rule of probability, but I want to understand the intuition behind it. Specifically, why does OR signifies addition in probability theory?

I came across the following question in a textbook (bear in mind that this is the only information given)-
There is a 50 percent chance of rain today. There is a 60 percent chance of rain tomorrow. There is a 30 percent chance that it will not rain either day. What is the chance that it will rain both today and tomorrow?
My instinct was to multiply the probabilities together for today and tomorrow to arrive at an answer of 30 percent. However, the answer given is 40 percent (based on taking the addition of the individual probabilities and subtracting their union). Can someone explain to me why the multiplication rule does not apply here? Does it have to do with independence? Bear in mind, I am trying to relearn probability theory from scratch. Thanks.

There's an 80% probability of a certain outcome, we get some new information that means that outcome is 4 times more likely to occur.
What's the new probability as a percentage and how do you work it out?
As I remember it the question was posed like so:

Suppose there's a student, Tom W, if you were asked to estimate the probability that Tom is a student of computer science. Without any other information you would only have the base rate to go by (percentage of total students enrolled on computer science) suppose this base rate is 80%.
Then you are given a description of Tom W's personality, suppose from this description you estimate that Tom W is 4 times more likely to be enrolled on computer science.
What is the new probability that Tom W is enrolled on computer science.

The answer given in the book is 94.1% but I couldn't work out how to calculate it!
Another example in the book is with a base rate of 3%, 4 times more likely than this is stated as 11%.