Why a both teams to score price moves
BTTS can be derived from two scoring rates. That is why it can move sharply while the match-result probabilities barely change.
Both teams to score is not a statement about whether a game will be entertaining. In a basic scoring model, 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 under an independent Poisson model, which lands at 53.2%.
Now suppose slower expected match conditions cut both rates by 0.3, to 1.3 and 0.8. The home-win probability in that model moves from 49.0% to 48.3%, while BTTS falls from 53.2% to 40.1%. The result view is almost unchanged, but the scoring view moves by more than 13 percentage points.
Why it decouples from the result
The home-win probability barely moved in that example because the difference between the two scoring rates stayed the same. BTTS moved a lot because both teams became more likely to draw a blank. The markets are related, but they do not respond equally to every change in the scoring rates.
The same distribution prices both, in the same way that every tennis market comes off one point model. Different assumptions can make the two markets appear inconsistent, so the margin and scoring model need checking before treating the gap as an error.
What this does not tell you
Independent Poisson scores are an approximation. Research on football scoring has found that scoring rates change with time and the current score, while the Dixon-Coles model adds a correction for dependence in low-scoring outcomes. The simple BTTS calculation is transparent, but it should not be treated as a complete match model.
Common questions
How is both teams to score priced?
In a basic independent Poisson model, it comes from each side's probability of scoring at least once. Richer models can adjust for dependence between the teams' scores.
Why can BTTS move differently from the match odds?
Because it depends on both sides scoring, not only on who wins. A change that lowers both scoring rates can move BTTS sharply while leaving the home-win probability close to where it was.
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.