Why a both teams to score price moves
BTTS looks like a mood about attacking football. It is two scoring rates multiplied, which is why it moves when nothing exciting happened.
Both teams to score is the most intuitive market on a football board and the one most often reasoned about wrongly. It is not a statement about whether the game will be entertaining. It is one minus the chance that at least one side draws a blank, and those blanks come from two separate scoring rates.
The arithmetic
If the home side's expected goals are 1.6 and the away side's 1.1, then under a standard scoring distribution the home side fails to score about 20% of the time and the away side about 33%. The chance that both score is roughly the product of the complements, which lands near 54%.
Now cut the away side's expectation to 0.8 because a first-choice striker is out. Their blank probability rises to about 45%, and BTTS falls to about 44%. Ten points of movement, from one absence, in a market that looks like it is about atmosphere.
Why it decouples from the result
The match odds barely moved in that example, because the home side was already favoured and became slightly more so. BTTS moved a lot. This is the general pattern: news that changes who wins moves the result market, and news that changes how much anybody scores moves this one.
The same distribution prices both, in the same way that every tennis market comes off one point model. A board where they disagree is a board where somebody priced one off the other with a rule of thumb.
What this does not tell you
The independence assumption is wrong and everybody knows it. A side chasing a game attacks more, so goals are correlated within a match in ways a product of two rates cannot capture. The approximation is decent in the middle of the distribution and poor at the extremes, which is where the interesting prices usually are.
Common questions
How is both teams to score priced?
From each side's probability of failing to score, derived from the same scoring distribution that prices the match odds. The two are combined rather than judged as a mood about the fixture.
Why does BTTS move when the match odds do not?
Because it depends on both sides scoring, not on who wins. News that makes one side weaker in attack moves BTTS without moving the result market much.
Is BTTS correlated with the total goals market?
Strongly, but not identically. A 3-0 win is over 2.5 goals and no BTTS, which is exactly the case where the two markets come apart.