Wandering Sammy

Let $X_i$ be the forward distance in meters that Sammy travels during the $i$ th minute and note that

$E(X_i) = \frac{1}{2}\cdot (1) + \frac{1}{3}\cdot (0) + \frac{1}{6} \cdot (-1) = \frac{1}{3}$

for all $i.$ Then, the forward distance Sammy has traveled after one hour is simply $X_1+X_2+\ldots+X_{60}.$ By linearity of expectation,

$E\left[\sum_{i=1}^{60}X_i\right] = \sum_{i=1}^{60} E\left[X_i\right] = 60 \cdot \frac{1}{3} = 20.$

Half of the time Sammy moves forward, so at the rate of each minute he will move 30m forward. He moves backward one-sixth of the time. So, 1/6 X 60 = 10. So distance travelled = 30 - 10 = 20m (NOTE: We can ignore the probability of staying where he is, as it does not affect the distance. It's in the left out 20 minutes as 1/3 X 60 = 20)

If Sammy spent half his time walking forwards, after 60 minutes he would have spend 60/2 = 30 minutes walking forwards. If he walks 1 metre per minute, in 30 minutes he would have walked 30 metres.

If Sammy spent a sixth of his time walking backwards, after 60 minutes he would have spend 60/6 = 10 minutes walking backwards. If he walks 1 metre per minute, in 10 minutes he would have walked back 10 metres.

If he walked 30 metres forward and 10 metres backwards, he would have walked 30 - 10 = 20 metres forwards in total.

Let $E[X_n]$ denote the expected distance travelled by Sammy in $n$ minutes.

Then $E[X_{n+1}]=\frac{1}{2}(E[X_n]+1) + \frac{1}{3}E[X_n] + \frac{1}{6}(E[X_n]-1)$ , based off the probabilities of moving $1$ meter forward, $0$ meters forward and $1$ meter backwards (equivalently $-1$ meters forward).

This simplifies to $E[X_{n+1}]=E[X_n]+\frac{1}{3}$ , or $E[X_{n}]=\frac{n}{3}+E[X_0]$ .

Since $E[X_0]=0$ , as Sammy can't have moved before the first second, $E[X_{60}]=\frac{60}{3}=\boxed{20}$

60 mins ( 1/2 - 1/3 + 1/6) = 20 m

We can ignore the probability of staying where he is, as it does not affect the distance. It's in the left out 20 minutes as 1/3 X 60 = 20

60 1/2+60 1/3+60 1/6 = 60/3 = 20 meters or the míddle value (I think is geometric mean) and multiply 60 1/3 = 20.