Let $P_0 \in [-0.5, 0.5]$ an initial point. We will iteratively draw values $x_i$ from a distribution $U(0, 1)$ If $i$ is odd, we add that number to $P_0$. If it is even, we subtract it from $P_0$.
If $P_0$ is shifted under $-0.5$ we terminate the process. Likewise, if it is shifted above $0.5$.
I want to compute the probability that the process is terminated by shifting $P_0$ above $0.5$. This event is a union of possible events, each with different probabilities. Namely, the ways this can happen are
- $x_1$ satisfies that $P_0 + x_1 > 0.5$.
- $x_1$ did not satisfy $P_0 + x_1 > 0.5$, $x_2$ did not satisfy $P_0 + x_1 -x_2 > 0.5$, $x^3$ satisfied $P_0 +x_1 -x_2 + x_3 > 0.5$.
- etc.
If we denote $X$ the event that the sum surpasses $0.5$, then
\begin{align} P(X) = P(P_0 + x_1 > 0.5) &+ P(P_0+x_1 < 0.5) \cdot P(P_0 + x_1 - x_2 > -0.5) \cdot P(P_0 + x_1 - x_2 + x_3 > 0.5) \\ &+ \ldots \end{align}
This seems overly complicated, since one would have to use some general convolution formula to manage the sum of random variables.
My question is whether it makes sense to use expected values to simplify the probabilities. For example, take the first case listed above and observe that $P(P_0 + x_1>0.5) = 1-(0.5 - P_0) = 0.5 -P_0$.
Now take the second case in the bullet points above. Assume the first draw did not satisfy $P_0 + x_1 >0.5$, the second did not satisfy $P_0 + x_1 - x_2 < 0.5$, the third did produce a sum above $0.5$.
The probability of $P_0 + x_1 < 0.5$ is $1-(0.5 -P_0)=0.5+P_0$, and expected value for $P_1 = P_0 + x_1$ is precisely $P_0 + 0.5$.
Then, can we say the probability of $P_2 = P_0 + x_1 - x_2 < 0.5$ is $P\big((P_0 + 0.5) - x_2 > -0.5\big)$? In other words, is it justified to substitue $P_0 + x_1$ with $P_0 + 0.5$, its expected value?
I'm very intuitively drawn to this idea because by removing the random variables $x_1, ..., x_n$ and using their expected values, we can express the probability (is it still a probability?) of our event only as a function of $P_0$, which is manageable. The problem, in its complete formulation (which I have not given), requires some objective function of $P_0$, since it asks what is the $P_0$ that gives equal chances to both terminating events: $P_x < -0.5$ and $P_x > 0.5$.
Note. I do not want a solution to the problem! Just concerned with the specific question on expected values.