Meteor probability question, assuming that at most one meteor can fall or more than one meteor can fall I know there have been similar questions on Math SE, but this has been bothering me for quite a while. Let's say you have the following question, but solve it with two different assumptions:

In the first hour, there is a $10\%$ chance that a meteor will fall into Earth. Assuming that the probability is uniform throughout (not too sure how to say this properly, but I'm trying to say that a meteor has equal chance of falling at any time), what is the probability that a meteor falls within the first $30$ minutes?

The two assumptions are:

*

*More than $1$ meteor can fall into Earth within that one hour

*At most $1$ meteor can fall into Earth within that one hour

Solving the problem with assumption 1:

Let the probability of a meteor falling within the first hour be $p$. Then, we can set up the following equation: $$\begin{align}P(\text{none fall in first 30 mins.})+P(\text{none fall in last 30 mins.)}&=P(\text{none fall in first hour})\\(1-p)^2&=0.9\\p&\approx5.13\%\end{align}$$

Solving with assumption 2:

Let the probability of a meteor falling within the first hour be $p$. Then, we can set up the following equation: $$\begin{align}P(\text{one falls in first 30 mins.})+P(\text{one falls in last 30 mins.)}-P(\text{one falls in each 30 minute intervals})&=P(\text{one falls in first hour})\\p+p-p^2&=0.1\\p&\approx5.13\%\end{align}$$

I feel like the two assumptions should produce different answers, but I can't pinpoint where the problem is.

Your answer re the 2nd assumption is wrong.
There is a $10\%$ chance of the meteor falling in 1 hour.
If this meteor falls in the 1st 30 minutes, then, by the 2nd assumption, it is impossible for a 2nd meteor to fall anytime within the hour.
So, the events that a meteor falls within the 1st 30 minutes and a meteor falls within the 2nd 30 minutes are :

*

*not independent, so $p(E_1, E_2) \neq p(E_1) \times p(E_2).$

*and actually disjoint, and therefore additive
So, there is a $5\%$ chance of the meteor falling within the 1st 30 minutes and a $5\%$ chance of the meteor falling within the 2nd 30 minutes.
