What is the probability that their first two goals are both 1 point goals?

Question

Consider a sports team who scores 1 point goals as a Poisson process of rate $\alpha$ and 3 point goals as an independent Poisson process of rate $\beta$. What is the probability that their their first two goals are both 1 point goals? What is the probability that the team scores two 1 point goals before they score two 3 point goals?

Anyone can help me to explain the structure of the question? This topic is new for me.

Answer 1

This question relies on the relationship between the Poisson and Exponential distributions (see this post). Here's one way to approach the first problem:

1. In order for the first two goals to be 1-point goals, at least two 1-point goals must occur before the first 3-point goal.
2. If you knew the time of the first 3-point goal, could you calculate the probability of two or more 1-point goals occuring before that?
3. How can you move from the probability of two 1-point goals occurring before the first 3-point goal, conditional on the time of the first 3-point goal, to the full probability of interest? (Hint: try to integrate using the probability density function for the time of the first 3-point goal).

Hope this helps!