# Science vs Vegans

One of my favourite podcasts just did an episode on veganism, and its great. Science Vs tackled three questions:

• Is going vegan better for the environment?
• Do you need milk for strong bones?
• If you don’t eat any meat at all, is it bad for you?

Listen here or wherever you get podcasts.

And if you liked that episode, check out:

Note: While the vegan episode shows that the science is firmly behind the diet, the same can’t be said for the other episodes above (sorry keto, detoxes, supplements, and even organic food). You might find your beliefs and lifestyle choices refuted by the science. That’s ok. Listen with an open mind, and challenge your own preconceived notions. Committing yourself to constantly learning and improving is probably the healthiest diet.

# Riddler Revisited

I’m taking a second crack at FiveThirtyEight’s Riddler problem. I thought I solved the problem two weeks ago, but my answer wasn’t quite right.

Here is this week’s problem:

There’s an airplane with 100 seats, and there are 100 ticketed passengers each with an assigned seat. They line up to board in some random order. However, the first person to board is the worst person alive, and just sits in a random seat, without even looking at his boarding pass. Each subsequent passenger sits in his or her own assigned seat if it’s empty, but sits in a random open seat if the assigned seat is occupied. What is the probability that you, the hundredth passenger to board, finds your seat unoccupied

I’m not even going to try and figure out the math for this puzzle. My statistics skills aren’t nearly strong enough. But my coding skills are. This is an easy problem to solve with a Monte Carlo simulation.

Here’s my python solution.

And the output charted.

The last passenger has a 50% chance of getting their assigned seat.

The number of passengers who get their assigned seats is distributed around 95, with a 1% chance of everyone getting their assigned seat (in the event the first passenger randomly chooses his/her own seat).

Interestingly, it doesn’t matter how many seats are on the plane. If there are 10 or 200 seats, the last passenger still has a 50% chance of getting their assigned seat. The second last passenger has  67% chance, and the 3rd last passenger has a 75% chance.

It’s easy to derive the equation for the Xth last passenger from this, even though I can’t figure out the math to prove it. The Xth last passenger has a X/X+1 chance of getting their assigned seat. So the 5th last passenger, has a 5/6 = 83.3% chance of getting their seat.

# Math Problems

I’ve noticed a few Grade 6 math problems on Facebook that I’m sure Einstein would be embarrassed to see his picture on. I’m not sure which is sadder, that people consider a problem given to 11-year olds genius-level, or that only 50% of adults got the answer right.

I’m much more interested in the problems that FiveThirtyEight has been posting for their Riddler series. This week the problem involves cars getting stuck in traffic. Thankfully, not something I normally have to deal with, but I think I figured out the answer.

The Problem:

There is a very long, straight highway with some number of cars (N) placed somewhere along it, randomly. The highway is only one lane, so the cars can’t pass each other. Each car is going in the same direction, and each driver has a distinct positive speed at which she prefers to travel. Each preferred speed is chosen at random. Each driver travels at her preferred speed unless she gets stuck behind a slower car, in which case she remains stuck behind the slower car. On average, how many groups of cars will eventually form? (A group is one or more cars travelling at the same speed.)

My solution:

f(N) <- average number of car groups if there are N cars
f(0) = 0
f(1) = 1

If we have 2 cars there is 50% chance of the first car being faster, which would create 2 groups; and 50% chance of the 2nd car being faster and merging into a single group.
f(2) = 0.5 * 2 + 0.5 * 1 = 1.5

More generally, if we just consider the first 2 cars. There’s a 50% chance of there being a solo lead car plus the groups that form behind it, and a 50% chance of the first car merging into the group behind it. We can define the average number of cars recursively, where:
f(N) = 0.5 * (1 + f(N-1)) + 0.5 * f(N-1)

that equation reduces to:
f(N) = 0.5 + f(N-1)

testing it out:
f(2) = 0.5 + f(1) = 1.5
f(3) = 0.5 + f(2) = 2
f(4) = 0.5 + f(3) = 2.5

Which is an arithmetic series that can be reduced to:
f(N) = 0.5 + 0.5 * N

or more succinctly:
f(N) = 0.5 * (N + 1) (for N >= 1)
f(N) = 0 (for N < 1)

# Nerdgasm

A Song of Ice and Fire
The TV series is good, but the books are even better. It’s difficult to describe what makes this series so amazing – but I like how the characters grow, evolve, and the suddenly die. George R. R. Martin has no problem killing off main characters, which was alarming at first, but as a reader it does keep things interesting. I just started book 4 – A Feast for Crows. If you’re looking for other good fantasy or science fiction books, check out this handy flowchart of NPR’s Top 100 list.

Elder Scrolls V: Skyrim
This game looks awesome! It comes out on 11/11/11. I’ve already pre-ordered my copy and am prepping for 3 days of non-stop dragon-slaying action on the November long weekend. I haven’t been this excited about a video game since Mass Effect.

Quantum Levitation
Hot damn that’s cool.