Logistic regression is a common classification method in supervised learning. Surprisingly , there are very few solutions for performing it and selecting variables in the presence of missing values. We develop a complete approach, including the estimation of parameters and variance of estimators, derivation of confidence intervals and a model selection procedure, for cases where the missing values can be anywhere in covariates. By well organizing different patterns of missingness in each observation , we propose a stochastic approximation version of the EM algorithm based on Metropolis-Hasting sampling, to perform statistical inference for logistic regression with incomplete data. We also tackle the problem of prediction for a new individual with missing values, which is never addressed. The methodology is computationally efficient, and its good coverage and variable selection properties are demonstrated in a simulation study where we contrast its performances to other methods. For instance, the popular multiple imputation by chained equation can lead to biased estimates while our method is unbiased. We then illustrate the method on a dataset of severely traumatized patients from Paris hospitals to predict the occurrence of hemorrhagic shock, a leading cause of early preventable death in severe trauma cases. The aim is to consolidate the current red flag procedure, a binary alert identifying patients with a high risk of severe hemorrhage. The methodology is implemented in the R package misaem.