Calamari Contest. This year's Calamari Contest ${?}^{11}$ involves an infinite bridge made of

blacked$-$out digital displays aligned end$-$to$-$end like concrete squares in a sidewalk. A

player, in attempting to cross the bridge, advances square$-$by$-$square. When a square

in a row is stepped on$,$ it either reveals a picture of fried calamari with probability $p$

$($and the player who stepped on it is eliminated$)$ or turns solid green, and is allowed to

advance to the next square. The player who goes the longest before being eliminated

wins.

$($a$)$ A player's lifetime is defined as the number of squares stepped on before calamari

is revealed. In expectation, what is a player's lifetime? Equivalently, what is $E[i]$

where $i$ is the return$-$value of walk $(p)$ in the following pseudocode?

from random import uniform

def is$\_$calamari$($p$)$:in the range $[0,1)$ uniformly at random.

return uniform$(0,1)$ p

def walk$($p$)$:

i $=0$

while not is$\_$calamari$($p$)$:

i$+=1$

return i

$($b$)$ Turns out we don't have the budget for an infinite bridge. What if the length of

the bridge is limited to $n0$ rows $($In other words, what is $E[i]$ where $i$ is the

return$-$value of walk$2(p,n)$ in the following pseudocode$)?$

def walk$2($p$,$ n$)$:

i $=0$

for $\_$ in range$($n$)$:

if is$\_$calamari$($p$)$:

break

i$+=1$

return i

$($c$)$ Wait a minute! If you did that right, the game got easier yet the life expectancy

decreased. This is correct. Explain why.