Note: This month, we kicked off the era of two level puzzles. Solving the first sets up key pieces needed to resolve the second. We hope this opens the puzzling up to more of you.
Congratulations to Purnima who solved Question 1 using a conditional probability approach and Paul D. who solved Question 2 using an approach similar to the one below!
A hot Mongolian folk band is playing tonight. The venue is a coffee house in the middle of a street that has 11 parking spots, as shown above. Each spot is open with probability p = 1/20.
If you approach from the right end of the street, what is the probability that the first open spot you encounter is zero or 1 spots away from the venue?
In order for a particular parking spot to be the first open one we see, two things must be true:
So, the probability that the spot to the right of the cafe is the first open one we see is
$$ p(1-p)^4, $$
since it has to be open (accounting for the leading $p$) and the four that come before it have to be occupied (accounting for the $(1-p)^4$).
So, the total probability that the first spot we see open is within one spot of the cafe is
$$ \boxed{P = \overbrace{p(1-p)^4}^\text{right} +\overbrace{p(1-p)^5}^\text{front}+\overbrace{p(1-p)^6}^\text{left}}, $$
which comes to $11.6\%$ when one in twenty spots is unoccupied.
The Mongolians are back and more popular than ever. They’re playing a bigger venue this time, and you want to park as close to the show as possible. The venue is in the middle of a very long one-way street—so you can’t return to a parking spot once you pass it. You also can’t see if a spot is open or not until you’re next to it and, as before, each spot is open with probability p = 1/20.
On average, how close can you park to the venue if you park optimally?
In the first part, we calculated the probability of a particular spot being the first open one we see. But now we need a parking strategy.