Probability of getting the gift I came across this problem: 

I and $m$ other people enter a plane which have $n$ seats. Each seat has a box kept over. It is given that there is only one box that contains a gift, the rest are just empty. What is the probability that I get the gift?

My question is, will the probability get affected due to other people? 
Consider the event that a person gets a gift, will this be an independent or dependent event?
 A: The probability that you get the box with the gift is $\dfrac{1}{n}$.
A: It depends on whether the distribution of the boxes to the $n$ seats is truly random and independent of passenger's preferences for particular seats. 
When the following model (M)  can be regarded as equivalent to the actual setup, then the probability that you will get the gift is ${1\over n}$.
(M) The $m$ passengers have taken seats by whichever procedure, and only then a number $N$ form $\{1,2,\ldots,n\}$ is drawn. If there is a passenger on seat $N$ he gets the gift; otherwise the gift is withdrawn.
A: There seems to be three cases. I use 1+m because the total number of people is you and m people.


*

*1 + m < n : Everyone gets a seat, but there's a chance that someone does not sit in the seat with the gift.

*1 + m = n : Everyone gets a seat and all seats are taken. Someone has to get the gift.

*1 + m > n : Not everyone gets a seat. Someone has to get the gift.


Let S be the event that you have a seat.
Let G be the event that your seat has the gift.
For all cases:
$P(you\ get\ the\ gift) = P(S\ and\ G) = P(S) * P(G)$
For case 1, you have:
$$
P(S) = \frac{n}{1 + m} = 1\ \ (due\ to\ pigeonhole\ principle)\\
P(G) = \frac{1}{n}\\
P(you\ get\ the\ gift) = 1 * \frac{1}{n} = \frac{1}{n}
$$
For case 2:
$$
P(S) = \frac{n}{1 + m} = \frac{n}{n} = 1\\
P(G) = \frac{1}{n}\\
P(you\ get\ the\ gift) = 1 * \frac{1}{n} = \frac{1}{n}
$$
For case 3:
$$
P(S) = \frac{n}{1 + m}\\
P(G) = \frac{1}{n}\\
P(you\ get\ the\ gift) = \frac{n}{1 + m} * \frac{1}{n} = \frac{1}{1 + m}
$$
Lastly, another way to think about it is as a competition. First, you determine the chance that you'll get to participate. Given that, next determine what are the chances that you'll win. Combine those results to reach a conclusion.
To answer your question, the last case does show that the number of people does affect your chances.
