Is the question I am trying to answer a conditional probability or an intersection of events?

Question

I am trying to understand the difference between conditional probability and an intersection of events better. The type of event I am studying is a gambling problem. Let's say that I have an hour at the casino to play a game N times in a row. And I have a bankroll S that can tolerate L losses. I am using a martingale strategy where I can only bet S times in a row before losing all my money. I want to win at least W times.

So I want to understand what are my chances of winning at least W times within those N times, while not losing L times in a row.

Is this a conditional probability?

i.e. am I asking given that I do not lose L in a row out of N times, what are my chances of winning at least W times out of N times?

Or is this is an intersection of events?

e.g. What are the chances that I win at least W times out of N times and I do not lose L times in a row out of N times?

Answer 1

It's an intersection. You want the probability that you win at least $W$ times and you don't lose $L$ times in a row. Conditional probability would arise if you asked for the probability of winning at least $W$ times if you don't lose at least $L$ times in a row.

One thing that's not clear to me from your description is what happens if you win $W$ times before you have played $N$ times. Do you quit at that point, or keep playing?

The game can be analyzed as a Markov chain with states $(w,l)$ where $w$ is the number of wins so far, and $l$ is the length of the current losing streak. You would have an absorbing state when you've played $N$ games, and another when you've lost $L$ in a row, and perhaps another when you've won $W$ games, if indeed you quit at that point.

Answer 2

The conditional probability of winning at least W times knowing that I do not lose L times during the game is calculated on the basis of the results during which I do not lose L times. So we don't deal with the possibilities of losing L times.

While the probability of the intersection will be calculated in relation to the set of results including those where I lose L times. This probability will be intuitively smaller than the conditional probability $P(W \cap L) = P_{L}(W).P(L) \leq P_{L}(W)$.