Scenario A: Let's say someone recieves money from his mother on 4th of December, and someone sends him money on an unknown day, what is the possibitlity for him to recieve the money on the same day?
Scenario B: Let's say someone recieves money from his mother on someday between 4th and 16th of December, and someone sends him money on an unknown day, what is the possibitlity for him to recieve the money on the same day?
Suppose that you can narrow down the days in which you can receive the present from this rando stranger to an interval of n days (which contain all days from the 4th to the thirteenth of december).
Then the probability that this person sends the person in th efourth of december. Assuming each day is equally likely is $\dfrac{1}{n}$ this is because there are n possible outcomes(each with probability $\dfrac{1}{n}$, and only one of them gets the present in the 4th.
Now:what is the probability they both receive the present on the same day if the mother can send the present on any day between the 4th and 16th. There are 13 days in this interval. Look at the pairs (x,y) where x is the day you get the present prom your mother and y the day you get the present from the father. There are $13n$ such pairs since there are 13 possible days the mother can send the present and $n$ days the stranger can send the present. So then by the rule of product there are 13n pairs. Out of these 13n pairs there are clearly only 13 where the person receives both at the same time. Namely the pairs $(4,4),(5,5),(6,6)\dots(13,13),(14,14)$ Assuming all pairs are equally likely there are 13 pairs that achieve what we want and 13n total pairs, so the probability is again $\dfrac{1}{n}$