Importance of Math: expected value, bluffing frequency, and randomization

Poker is a mathematical game. Before you get me wrong, I'm not a "math player." That is, I'm not exceptional at math nor would I fit the stereotypical "math player" and I understand that poker is a people's game. What I mean when I say that poker is a mathematical game is that the right play is deeply rooted in math and understanding the person's tendencies helps plug you it into a mathematical equation to find the right play. I know I'm going in circles so I'll explain in examples.

For example, the right frequency to bluff in a situation is determined based on the likelihood your opponent will fold in that particular situation. In limit, for example, when all the cards are out, how do you determine whether you should bluff on the river? If you can figure out that the pot is offering a caller 5:1 odds to call, but you know that they will fold the best hand 1 in six times, then bluffing now becomes mathematically correct.

For example in NL, the right amount to bet is the amount that maximizes your expected value. In order to know how to maximize your expected value, you must be able to accurately assess how much a person is likely to call based on their likely range of holdings. If you put someone on a flush draw, what is the maximum that person will call to chase their draw while offered the wrong odds? The difficulty of maximizing expected value is when you do not know what cards your opponent likely hold.

Being able to accurately gauge what a person is likely to be playing (reading a player) helps you plug the right percentages into the mathematical formulas that allow you to play very optimally. How do you get good at reading a player's likely cards? Observation. You must watch them in many different situations so that you may predict accurately what they are doing in a situation with you. For example, if you wanted to predict what your girlfriend/boyfriend or mom/dad would do in a certain situation, you might be right very often because you've observed them in many many different situations and have an understanding of what they are thinking. People's betting patterns, frequency of raising versus folding, number of pots they play, how often they betting or raising, how often they calling or folding all give information on how someone plays even if you never get to see their cards face up out of the muck.

Once you get good at that, you can beat many medium stake games. But in order to win at the next level, you must also be able to deceive opponents who are actively observing you and assessing your play. You do this by choosing a mixed strategy. Quite simply, you randomize your plays so that even if they are reading you, they'd fail because even you do not know what you will do since you have randomized your play. In a lower stake game, its almost unneccessary to randomize since playing straightforward poker will usually win since your opponent's are not thinking about what you have. In higher stake games, your opponents are spending considerable mental resources on figuring out how you think, how you play, what strategies you use in certain situations, or what your betting patterns say about your game. In that case, you randomize and the percentage you chose one action over another is dependent on many variables in which you can apply in a mathematical formula to find the exact right percentage.