What is mathematical relationship to be able to determine number of times higher an even occurs in a logistic regression coefficient I'm working through material on logistic regression and I'm trying to understand how the mathematical relationship works to be able to make a certain claim. This excerpt comes from Beyond Multiple Linear Regression - Roback, Legler - pg. 221

So the coefficient being discussed here is the intercept which after we exponentiate we get the value of $0.0035$. One way to interpret this would be to say "the odds a foul is called on the home team is $0.9966$ times less likely to occur compared to the visiting team when there are an equal number of fouls." The converse of this is what is said in the text "the probability that a foul is called on the visiting team is $1/0.0035 = 289$ times higher......". In this case the response variable is binary. But I want to know what is the mathematical relationship from which they obtain the $289$ times from?
So using the base idea of our logistic regression model $\log(\frac{p}{1-p}) = \beta_0 + \beta_1X$, where $\log(\frac{p}{1-p}) = \log(\text{odds})$.  after exponentiating and such we will have the relationship
$$\frac{p}{1-p} = \beta_0$$.
So I'm assuming once we've run the regression in this particular case we have some sort of setup where:
$$0.0035 = \beta_0 = \frac{p}{1-p}$$
So how does $289$ come about? It is the "multiplicative factor", but I know it is not a probability and I can't solve for it as one in this situation.

The authors have explained this part in a very roundabout way, but I'll show you what they're doing.  I'm going to use hats over all my variables, since these are estimates being made from data.  After exponentiating both sides, you should have:
$$\frac{\hat{p}}{1-\hat{p}} = e^{\hat{\beta}_0}.$$
So the authors are saying that they have an estimated probability ratio:
$$\frac{1-\hat{p}}{\hat{p}} = e^{-\hat{\beta}_0} = \frac{1}{e^{\hat{\beta}_0}} = \frac{1}{e^{-5.67}} \approx \frac{1}{0.0035} \approx 289.$$
(Actually, we have $\exp(5.67) \approx 290$ to the nearest integer, so their rounding also seems a bit odd.  Perhaps they actually have more decimal places for their estimated coefficients than they are showing here.)
