If a football team has to leave a tournament after they lose. & in each match they have a probability $q$ of losing. Assuming each match is independent. IF they play X total matches.
Am I right in saying that the distribution of X is the following:
$X\sim Geo(q)$ ?
I have the follow up questions then:
- Show that $\mathbb{P}(X\gt x) = (1-q)^{x}$ for $x \in \lbrace 0,1,2,3,\dots\rbrace.$ Interpret this result
- Find a formula for the probability that X is even.
For the first part I know that the probability that the boxer wins or draws a match is $1-q$. So for him to win his first x matches he would need to win x times with probability $(1-q)$ so we get $(1-q)^x$
For the second part I'm not sure what to do. I've considered using the sum of a geometric series formula but I'm unsure on whether this is the correct way to approach the question. What would the initial value be for $a$? I think the common ratio would be $1-q$ but again I'm not sure