# Combining probabilities in case of incomplete information

The following is a use case from the hospitality industry.

Based on historical data it is known that bookings for rooms are often cancelled. Apart from other factors cancellation depends upon the number of rooms books under one order and how many days before check-in the booking was done.

It was observed that when a customer books $x$ rooms in one single order, the probability that a booking was not cancelled is given by:

$$p(x) = ax^{-b}$$

where $a$ and $b$ are suitable positive constants.

Similarly when the customer booked rooms $y$ days before the check-in date, the probability that the booking was not cancelled is given by:

$$p(y) = my - c$$

where $m$ and $c$ are suitable positive constants.

The two calculations were done independently of each other, i.e. in the first case the analysis was done based only on the number of rooms and not on how many days the planned check-in date is prior to the booking date. Similarly, in the second case, the number of rooms was not considered.

Therefore we do not know the conditional probabilities $p(x|y)$ or $p(y|x)$.

Question: Based on only this much information is it possible to come up with an equation which gives the probability $p(x,y)$ that the booking was not cancelled if the booking had $x$ rooms and was booked $y$ before the planned check-in date?

If the original models did not consider conditional probabilities you are only left with the option of treating both these probabilities as independent. $P(not cancel) = 1 - P(cancel)$.
Here $P(cancel)$ would be $(1 - ax^{-b})\times(1-my +c)$