I tried asking r/learnmath twice and got no replies unfortunately, so I'm going to repost it here:
These are example proofs from Proofs by Jay Cummings, so I'm not sure if I need to 'show work' because it's not an exercise, I just want to understand them. Here's a link to pictures of them for reference:
My probability is very flaky, because when I was a younger math nerd I ignored studying it because it didn't feel rigorous enough, which is probably the main source of my problems right now. In the first one, I'm mainly struggling with understanding how the probability that "at least one of these $K_t$ is monochromatic is certainly no more than...". I understand why the probability of a single $K_t$ being monochromatic is $2(\frac{1}{2})^{t\choose 2}$, but why does multiplying this by $n\choose t$ only give an upper bound for the probability of at least one monochromatic $K_t$ and not equality? Under what conditions would the actual probability of a monochromatic $K_t$ differ from ${n\choose t}\cdot 2(\frac{1}{2})^{t\choose 2}$? I'd imagine it's probably like an inclusion-exclusion thing, but I'm not sure. Everything afterwards or before I'm good on
For the second proof, I'm getting lost in the last paragraph. After "Fix any s in S, ..." he says "since each $a_i\in \{1,2,...,p-1\}$," which I'm struggling with because, at the beginning, the only restrictions for p were "$p>2a_{max}$" and $p=3k+2$ for some integer k, so wouldn't each $a_i \in \{1,2,...,\lfloor p/2\rfloor\}$? Why is $a_i$ an integer from 1 to p-1, in other words. Secondly, if $a_i\in\{1,2,...,p-1\}$, why is there "precisely one $x\in\{1,2,...,p-1\}$ for which $a_i x=s$." I'm not sure what property guarantees that an x exists such that $a_i x=s$, or even why it's unique. After and before that I'm good, but I'm just so completely lost at this part
As a side note, what would be a good introductory book into probability for someone who's already fairly adept at other areas of math (or just a good introduction to probability, I just don't want to be so oversimplified because it's intended for high-school students and I walk away knowing barely little more).
Let me answer your concerns about the two proofs.
For proof 1, that "multiplying by ${n\choose t}$ only gives an upper bound for the probability of at least one monochromatic $K_t$" is because you are counting with repetitions.
More precisely, let $$ C=\{2-\text{colorings of the edges of }K_n\}. $$ Let $K_t\subset K_n$ be one complete subgraph with $t$ vertices, and $K_t'$ be another subgraph with $t$ vertices. Let $M_t\subset C$ be the subset of $2-$colorings of $K_n$ that is monochromatic on $K_t$, and $M_t'$ be the corresponding set for $K_t'$.
The claim is that for any $K_t$ and $K_t'$, we have $$ M_t\cap M_t'\neq \emptyset. $$ Indeed, if you color the whole $K_t$ by one color, say red, then this element of $C$ belongs to $M_t\cap M_t'$, so it is counted more than once. So this upper bound is never a precise value.
Here is a more concrete example. Let $n=3$, $t=2$. Then we are coloring the 3 edges of $K_3$, and see the probability that there exists one $K_2$ that is monochromatic. Since each $K_2$ has only one edge, it is for sure monochromatic. So this probability is 1.
But the formula $$ {n\choose t}2\big(\frac{1}{2}\big)^{t\choose 2}=3\cdot 2\cdot \big(\frac 1 2\big)^1 = 3, $$ so it is only an upper bound.
About proof 2, the book says "since each $a_i\in \{1, 2,\dots,p-1\}$ because he only needs this condition. As you say, $a_i$ actually belongs to the subset $\{1, 2,\dots,\lfloor p/2\rfloor\}$, but he does not actually need this more detailed information.
As to why "precisely one $x\in \{1, 2,\dots,p-1\}$ for which $a_i x=s$", note that he is counting mod $p$, so the statement is precisely $$ \forall a_i, s\in \{1, 2,\dots, p-1\}, \exists ! x\in \{1,2,\dots, p-1\} \text{ s.t. }a_i x=s\text{ mod }p. $$ This follows from that $\{1,2,\dots,p-1\}=({\mathbb Z}/p{\mathbb Z})^*$ is the group of units (that is, invertible elements) mod $p$ under multiplication. Therefore the solution to $a_i x=s \text{ mod }p$ is uniquely $x=a_i^{-1} s$.
(A little more on the math side: Note that $({\mathbb Z}/p{\mathbb Z})^*$ is actually a cyclic group of order $p-1$, that is, there is a primitive element which is a generator. For example, in $({\mathbb Z}/7{\mathbb Z})^*$, $3$ is a generator.)