How do you calculate expected value in this game of Heads or Tails?

Question

Say you're playing a game. Each win = 1 point. Each loss = -1 point. Win rate is 50%. How do you calculate the expected value of the number of games needed to get to 5 points?

Answer 1

This question can be reformulated as a simple one dimensional random walk that steps up or down with probability $1/2$ and $\tau$ is the waiting time to reach 5 points (also known as the first passage time), so you are interested in $E[\tau]$. By setting up a recursion, it can be shown that $E[\tau]=\infty$. This does not imply the walk never hits 5, just that the expectation diverges.

Answer 2

The poster asks how to calculate the expected value. As others have suggested, one must know the specific sequence of the events to understand the sum of the events. The closest we can come to calculating the result would be the binomial distribution, as others have already demonstrated.

I was interested in the question and wanted to provide some sense of the number of turns, as originally asked. I used a Monte Carlo method, similar to Antoni's solution. In R code:

eq5 <- 0
for (j in 1:100000){
  sum <- 0
  trial <- 0
  turn <- 0

  for (i in 1:100){
    trial[i] <- sample( c(-1,1) ,1 , replace=TRUE)
    sum <- cumsum(trial)
    turn <- i
    if(sum[i] ==5) break
  }
  eq5[j] <- turn
}

hist(eq5, xlim=c(0,100), breaks=c(0:100), xaxt="n")
axis(side=1, at=c(0:100))
table(eq5)

Note: I ran only 100 turns causing a truncation of the very long right-sided tail.

The most common number of turns where a value of 5 occurs is shared between the 7th and 9th turn.