# A Mathematical Jaunt via San Mateo Parking Meters

Theeee other day some colleagues and I drove into downtown San Mateo for lunch. An interesting decision had to be made: how many quarters to pay the meter.

The price for parking is \$0.25 per half hour. Your penny-pinching instincts suggest to only put in 2 quarters. Let’s say that a parking ticket for going over the meter is \$40.

## Breaking Down the Scenario

Let’s apply a little reasoning and mathematics to this, situation and see what’s the right amount of quarters to put into the meter.

There are 5 possible outcomes:

1. pay \$0.50, go under one hour, no ticket: total price \$0.50
2. pay \$0.50, go over one hour, no ticket: total price \$0.50
3. pay \$0.50, go over one hour, get ticket: total price \$40.50
4. pay \$0.75, go under one hour, no ticket: total price \$0.75
5. pay \$0.75, go over one hour, no ticket: total price \$0.75

## Conservative Guesstimates

Now the fun part. Let’s assign rough guesstimates of likelihood for each outcome.

Let’s be generous and say that you only go over one hour 20% of the time, and only 20% of the times you go over do you get a ticket.

Let’s also say that you pay \$0.50 half the time and \$0.75 half the time.

1. 40% probability x \$0.50 = \$0.20
2. 8% probability x \$0.50 = \$0.04
3. 2% probability x \$40.50 = \$0.81
4. 40% probability x \$0.75 = \$0.30
5. 10% probability x \$0.75 = \$0.075

Computed average cost: \$1.425

Even at the conservative rate of ticketing of 20% of time you go over an hour, you still increase your expected rate of pay to almost double, by choosing to pay \$0.50 half the time!

## With Vigilant Meter Maids

If instead you think you’ll get a ticket 75% of the time you go over, the average cost per park becomes a whopping \$3.625

1. 40% probability x \$0.50 = \$0.20
2. 2.5% probability x \$0.50 = \$0.0125
3. 7.5% probability x \$40.50 = \$3.0375
4. 40% probability x \$0.75 = \$0.30
5. 10% probability x \$0.75 = \$0.075

Computed average cost: \$3.625

So it looks like in general it pays to be safe and pay the extra quarter when parking in downtown San Mateo.

## Further Precision

We could expand this further, by considering likelihoods of lunch going over 1.5 hours. I didn’t because I think there’s diminishing returns on investment because the odds of going over 1.5 hours are probably only like 20% of 20% => 4%… I’ll leave that as an exercise for the reader :)