If I know the mean, standard deviation, and size of sample A and sample B,...

Explanation:
Let us start with samples with unequal sizes.
Let the mean, standard deviation and size of sample A be ${\displaystyle {\bar{X}}_A,S_A}$ and ${\displaystyle n_A}$
and mean, standard deviation and size of sample B be ${\displaystyle {\bar{X}}_B,S_B}$ and ${\displaystyle n_B}$ respectively.
Then mean of comibined sample ${\displaystyle \bar{X}}$ is given by
${\displaystyle \bar{X}=\frac{n_A{\bar{X}}_A+n_B{\bar{X}}_B}{n_A+n_B}}$
and Standard Deviation of combined sample S is
${\displaystyle S=\frac{n_A(S_A^2+{({\bar{X}}_A-\bar{X})}^2)+n_B(S_B^2+{({\bar{X}}_B-\bar{X})}^2)}{n_A+n_B}}$
Note that it is important to work out mean of combined sample first, as to is used to calculate Standard Deviation of combined sample.
If the sample sizes are equal then the above reduces to
${\displaystyle \bar{X}=\frac{X_A+X_B}{2}}$ and
${\displaystyle S=\frac{(S_A^2+{({\bar{X}}_A-\bar{X})}^2)+(S_B^2+{({\bar{X}}_B-\bar{X})}^2)}{2}}$