Checking Whether Given Logistic Regression Classifier classifies data A bank wants to decide whether a customer can be given a loan,
  based on two features related to (i) the monthly salary of the customer, and (ii) his/her
  account balance. For simplicity, we model the two features with two binary variables $X1$, $X2$
  and the class $Y$ (all of which can be either 0 or 1). $Y=1$ indicates that the customer can be
  given loan, and Y=0 indicates otherwise.
  Consider the following dataset having four instances:
  ($X1 = 0$, $X2 = 0$, $Y = 0$)
  ($X1 = 0$, $X2 = 1$, $Y = 0$)
  ($X1 = 1$, $X2 = 0$, $Y = 0$)
  ($X1 = 1$, $X2 = 1$, $Y = 1$)
  Can there be any logistic regression classifier using X1 and X2 as features, that can perfectly classify the given data?  


The approach followed in the question was to calculate respective probabilities for Y=0 and Y=1 respectively.  The value of $p$ obtained was $0.25$ and $(1-p)$ as $0.75$.The $log(p/1-p)$ is coming as negative. But I am stuck at this place as to which parameter to be calculated which decides that logistic regression would classify/not classify the data ?
check it

import keras
from keras.layers import *

X = np.array([[0,0], [0,1], [1,0], [1,1]])
Y = np.array([[0], [0], [0], [1]])

input = Input(shape=(2,))
output = Dense(1, activation='sigmoid')(input)
model = keras.Model(input, output)

model.compile(keras.optimizers.Adam(1e0), 'binary_crossentropy', metrics=['acc'])
model.fit(X, Y, epochs=10, batch_size=4, verbose=1)


which produces 

Epoch 1/10
4/4 [==============================] - 0s 52ms/step - loss: 0.7503 - acc: 0.7500
Epoch 2/10
4/4 [==============================] - 0s 817us/step - loss: 0.5142 - acc: 0.7500
Epoch 3/10
4/4 [==============================] - 0s 732us/step - loss: 0.4353 - acc: 0.7500
Epoch 4/10
4/4 [==============================] - 0s 694us/step - loss: 0.3413 - acc: 1.0000
Epoch 5/10
4/4 [==============================] - 0s 633us/step - loss: 0.2817 - acc: 1.0000
Epoch 6/10
4/4 [==============================] - 0s 679us/step - loss: 0.2299 - acc: 1.0000
Epoch 7/10
4/4 [==============================] - 0s 672us/step - loss: 0.1769 - acc: 1.0000
Epoch 8/10
4/4 [==============================] - 0s 721us/step - loss: 0.1412 - acc: 1.0000
Epoch 9/10
4/4 [==============================] - 0s 694us/step - loss: 0.1193 - acc: 1.0000
Epoch 10/10
4/4 [==============================] - 0s 716us/step - loss: 0.1015 - acc: 1.0000


...so yes, you can

Also note you calculated marginal probabilities, here you want them conditioned on the input variables to actually solve for the parameters