@Maarten Buis has done an excellent job explaining your logistic regression questions, but I figured I'd supplement his answer with some sample R code so you can see the calculations performed in a statistical analysis program.
In the first part of your question, you asked about the computation of:
$$ log( odds(B=1|A) = \beta_0 + \beta_1 x_1 $$$$ log( odds(B=1|A) = \beta_0 + \beta_1 A $$
We can get this model ( log-odds of $B=1$ given $A$) as follows:
> A<-c(rep(0,44+443), rep(1, 27+95))
> B<-c(rep(1, 44), rep(0, 443), rep(1, 27), rep(0, 95))
>
> #view dummy table in table form
> table(B,A)
A
B 0 1
0 443 95
1 44 27
>
> #run log(odds(B=1|A))
> mylogit_BgivenA <- glm(B ~ A, family = "binomial")
> mylogit_BgivenA
Call: glm(formula = B ~ A, family = "binomial")
Coefficients:
(Intercept) A
-2.309 1.051
You can see from the coefficients the -2.309 here, corresponds to the log odds of B being equal to 1 given, all the values of A are equal to 0. This also corresponds to Maarten's approximate value of -2.3.
Moving on to your second question, we can find the probabilities and log odds of A by regressing A onto a constant value of 1:
> mylogit_Aonly <- glm(A ~1, family = "binomial")
>
> mylogit_Aonly
Call: glm(formula = A ~ 1, family = "binomial")
Coefficients:
(Intercept)
-1.384
>
> exp_b0<-exp(-1.384)
> #probability of event A given no explanatory variables
> #calculate the probability of p_A
> exp_b0/(1+exp_b0)
[1] 0.2003674
Lastly, note that $log(odds(B=1|A)) \ne log(odds(A=1|B))$. You can verify this by running another logistic regression and comparing the results with the first output above:
> #run log(odds(A=1|B))
> mylogit_AgivenB <- glm(A ~ B, family = "binomial")
> mylogit_AgivenB
Call: glm(formula = A ~ B, family = "binomial")
Coefficients:
(Intercept) B
-1.540 1.051
Note that the value of the intercept (-1.54) is different than that calculated in the first output shown (-2.309).