We are tasked with finding the probability $P(S_{10} > 0)$, where $S_n$ represents a simple random walk starting at $S_0 = 0$, and each step $X_i$ is either +1 or -1 with equal probability 0.5. That is, for each $i$, we have:

$P(X_i = +1) = P(X_i = -1) = 0.5$

Approach

The random walk $S_n$ is the sum of independent identically distributed (i.i.d.) random variables $X_1, X_2, \ldots, X_n$, such that:

$S_n = X_1 + X_2 + \dots + X_n$

For $S_{10}$, we need to find the probability that the sum of 10 steps is positive:

$P(S_{10} > 0)$

Symmetry of the Random Walk

This problem can be simplified by using the symmetry of the random walk. Each $X_i$ has an equal chance of being +1 or -1, so after 10 steps, the possible values of $S_{10}$ range from -10 to +10. Specifically, $S_{10}$ will be positive if there are more +1's than -1's in the 10 steps.

Thus, we are looking for the probability that more than half of the steps are +1. In other words, we need to compute the probability that the number of +1's, say $k$, satisfies $k > 5$.

Binomial Distribution

The number of +1's in 10 steps follows a binomial distribution with parameters $n = 10$ and $p = 0.5$. The probability mass function of the binomial distribution is:

$P(\text{Number of +1's} = k) = \binom{10}{k} \left( \frac{1}{2} \right)^{10}$

We are interested in the probability that $k > 5$, i.e., $k = 6, 7, 8, 9, 10$:

$P(S_{10} > 0) = P(k > 5) = \sum_{k=6}^{10} \binom{10}{k} \left( \frac{1}{2} \right)^{10}$

Calculation

$P(S_{10} > 0) = \frac{1}{2^{10}} \left( \binom{10}{6} + \binom{10}{7} + \binom{10}{8} + \binom{10}{9} + \binom{10}{10} \right)$

The binomial coefficients are:

$\binom{10}{6} = 210, \quad \binom{10}{7} = 120, \quad \binom{10}{8} = 45, \quad \binom{10}{9} = 10, \quad \binom{10}{10} = 1$

So:

$P(S_{10} > 0) = \frac{1}{1024} (210 + 120 + 45 + 10 + 1) = \frac{1}{1024} \times 386 = \frac{386}{1024}$

Simplifying:

$P(S_{10} > 0) = 0.377$

Thus, the probability that $S_{10} > 0$ is approximately 0.377.

Would you like more details or clarifications on any part of this?

Here are five questions you might find interesting based on this concept:

1. What is the probability that $S_{10} = 0$?
2. How does the probability $P(S_n > 0)$ behave as $n \to \infty$?
3. What is the probability that $S_6$ is exactly equal to 2?
4. How would the calculation change if $P(X_i = +1) \neq P(X_i = -1)$?
5. How can we generalize this to a two-dimensional random walk?

Tip: The binomial distribution is key in calculating probabilities for random walks, especially for counting the number of steps in one direction versus another.