*Spending couple of weeks down the south coast of New South Wales, spotting dolphins and echidnas, has slowed down my blogging. Fortunately, regular contributor James Glover has once more come to the rescue with a guest post. This time his topic is wheelchairs and air-travel.*

Perhaps you’ve heard of a recent court case in which a wheelchair user, Sheila King, took Jetstar to court (and lost) on the basis of the Disabilities Discrimination Act? If you are a wheelchair user and you book a flight on one of our airline carriers then a fairly obvious thing won’t happen. Unlike say a bus you won’t be able to board the aircraft in your chair and be strapped in for the journey. What actually happens is that when making the booking you tick a box (or tell the booker on the phone) that you are in a wheelchair. If there are seats available for wheelies when you get to the airport you will give up your chair and be made to use a specially designed “wheelchair” (its a chair, it has wheels) that is designed to be fit the narrow corridor of most planes which I am sure you are aware of – their narrowness, for you, only apparent when the person ahead of you is blocking the aisle loading 3 pieces of carry on luggage into the overhead lockers while chatting to their new friends in the seat they are meant to occupy. We all suffer this situation. These “wheelchairs” are not designed to be used without help, they are more like children’s toy carts and cannot be operated by the user as the wheels are very small and low down. For a wheelchair user to be taken out of their wheelchair in a public place can be quite discombobulating. Many wheelchair users develop a personal relationship with their chair – it is after all a place you spend many of your waking hours.

Digression. The very first time I was in a wheelchair outside the confines of a hospital ward (it was a hospital wheelchair but is the exact same model I now own, like I said it is personal) I was being pushed by none other than the proprietor of this very website! Without going into the details let’s just say it was a pretty dramatic event and we both learned a valuable lesson in wheelchair use and the wheelchair repair workshop at the hospital was kept busy. But I digress.

So here is the thing. According to Google about 1% of the population uses wheelchairs. And a Jetstar plane has about 200 seats so they expect to get about 2 wheelchair users on average per flight. So what is the problem with only allowing this same number on each flight, as some airlines do? Well the problem is that statistically wheelchair users don’t travel in pairs and sometimes there will be less than 2 users and sometimes there will be more. Just as if you toss 10 coins sometimes there will be fewer than 5 heads (the average or expected number) and sometimes there will be more. Only on average will there be 5. In fact it is a simple problem to work out the probability of there being, say, *n* wheelchair users, given the average of 1% on a 200 seat plane. This is called the Binomial Distribution. If you have access to Excel then the function Binomdist(n,200,1%) will tell you this probability. Before I give you some numbers I admit that the overall population average may not be the same as the average flying on planes. It may be less than 1% due to wheelchair users being put off flying. But maybe on some routes it is higher: but I am guessing the annual “snowbird” migration of retired people from the northern United States to Florida at the start of Winter would track above the 1% rate.

So here are the Binomial probability figures.

Count | Probability |
---|---|

0 | 13% |

1 | 27% |

2 | 27% |

3 | 18% |

4 | 9% |

5 | 4% |

6+ | 2% |

**Binomial Probabilities (N=200, p=1%)**

For example, assuming a 1% chance of any given passenger a 200 hundred seat plane being in a wheelchair, the probability that there will be exactly 4 wheelchair passengers wanting seats is 9%. To work out the probability of a passenger being denied a seat on their preferred flight, we will assume that we’re dealing with an airline where more than two wheelchair passenges book on a flight, then at least on passenger will have to change their travel plans. From the table above, the chance of the flight only having 0, 1 or 2 wheelchair passengers totals 68%, so there’s a 32% chance that there will be at least one wheelchair passenger who cannot fly. For any one wheelchair passenger, there is a (*n*-1)/(*n*+1*)* chance of being bumped if *n* other wheelchair passengers book on the flight. Weighting that by the probably that there are *n* passengers and adding it up for all *n*>1 gives a probability of 27%. As a frequent flyer in a wheelchair, you can expect to miss out on a seat quite regularly! [Note: these calculations have been updated: the editor’s “corrections” were undone. Ed.]

I am quite fortunate now that I no longer need to travel in my wheelchair. But as I still use a walking stick I wait for everyone else to get off the plane. You sit there, looking behind you to see if everyone else has left. But there are always these strange people who seem to sit there at the back of the plane and wait for 10 minutes or more, after everyone has disembarked, before even moving. You wonder why the airline staff don’t just hurry them off? I assume they aren’t disabled because they are sitting at the back of the plane. If airlines really had a problem with the extra time that getting wheelies off the plane then they could make this up by just moving these people along.

When I first read about this case my initial response was that being disabled and traveling is a bit of challenge anyway and you just get on with it. But the more I thought about it I wondered if the airlines just took it for granted that wheelchair users would change their plans to fit in with the rules. I am glad Sheila King took the issue up!

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MagpieHey Jimbo!

I don’t suppose those limitations should affect you when you travel to the Davos 2012 meeting this week. ;-D

If you need a body-guard, just drop me a line!

ZebraIn my original version of this post I asserted that the probability of getting bumped in a wheelchair was 26%. The Mule (unilaterally)changed this to 14% calculated as the probability of there being n-wheelchair users on the plane and that for n>2, there would (n-2) of them bumped so prob=Sum((n-2)/n*B(n,N,p))=14% which for N=200, p=1% the sum is correct. My figure was based on calculating the expected number of people being bumped divided by the expected number of wheelchair users. The latter is just pN while the former is Sum((n-2)*B(n,N,p)) for N>2 so prob=Sum((n-2)*B(n,N,p))/pN=26%. To reconcile them note that the Mule hasn’t taken into account that the probability of being a wheelchair user on a plane with say 3 wheelchair users isnt B(3,N,p) (the prob of a plane having 3 wheelchair users) but has to also take into account that there are, for example, 3/2 times as many chances of being in that cohort as say a cohort with the same binomial prob but only 2 members. The correct formula is then: prob=Sum((n-2)/n*B(n,N,p)*(n/Np))=Sum((n-2)*B(n,N,p)/pN) which is my first formula. In order to be sure I simulated this process and got the same answer.

Zebra@magpie – bodyguard? As in Kevin Costner and Whitney Huston?

MagpieOf course!

But you better not sing to me or come up with funny ideas (not that there’s anything wrong with that: just that I’m too old to change habits).

Zebra@magpie – if this doesnt make your blood boil and break into “I Will Always Love You Poor Billionaires” then nothing will. http://www.bloomberg.com/news/2012-01-24/billionaires-occupy-davos-as-0-01-bemoan-economic-inequalities.html

MagpieI suspect one guy (the one speaking more like C. Montgomery Burns) did say the truth: “it’s all for the cameras”.

In any case, seeing is believing.

zebraIn case anyone is wondering why I went to the extreme of simulating this situation to prove (to myself) I was right I should point out that in the 14 years or so of knowing The Mule I have only ever bested him in an argument once (about the definition of the word “Canonical” of all things). So I wanted to be confident before calling him on it. I am looking forward to an admission of him being wrong. If so I will copy it and frame it and put it on my wall. And get him to sign it. With witnesses. Oh yes I will.

MagpieActually, I was wondering about that…

So, Stubborn, what do you have to say in your defense?

Stubborn MuleHaven;t checked it yet, but in my defence, I did call the Zebra to discuss the calculation…the response what something along the lines of “change whatever you like”.

Stubborn MuleP.S. @Zebra don’t forget you can edit your own posts too!

ZebraIn my defense (note spelling) there are very few people who I am reluctant to argue against on the spot, if they suggest I might be wrong, before going back to recheck my working. The Mule is one of them. Possibly the only one.

Zebra@mule – reframing the argument?

Stubborn MuleOK, I’ve thought about it again: my initial attempt was wrong: I didn’t condition on the fact that the given wheelchair passenger was on the plane. I’ve redone it with

nrepresenting the number ofotherwheelchair passengers on the plane (n=0, 1,…). This probably ends up being equivalent to your approach.zebrathanks!

MagpieMr. Zebrovsky,

Our friend may be stubborn, but he’s fair, eventually… ;D

KenThe assumption that the proportion of wheelchair users in the population and the number seeking to fly would almost certainly be incorrect. Socioeconomic factors and other illnesses would reduce the probability that wheelchair users would fly. It is surprising that the rate of refusal wasn’t raised in the court case.

Zebra@ken – I generally agree – the inconvenience alone would deter many from frequent air travel. On the other hand given that on average people in wheelchairs are economically worse off they may be more inclined to travel the budget airlines and this may have pushed then numbers back up towards the average.

evoMule – please consider retitling said article to

“Have wheelchair, will travel…probability”

WisdomToothFirst, a note from our editor-in-chief: “For some as yet undiagnosed reason, the Stubborn Mule’s email subscription service re-sent a post from January. I will be digging into the cause and hope it doesn’t happen again.” Well, I kinda liked blissfully reading it thinking it was brand new. It’s still pretty current with NDIS and the topic below being talked about.

Now… “If you can’t solve it, upscope it.” – (don’t know who said it, but someone smart must have… otherwise I’ll take the credit ;))

N2S, this is a sensitive topic, of course. (Btw, disclaimer: I am one of those ‘strange people’ lingering about at the back of the plane, aloof in my reading of something more interesting than standing in line with the other sheeple to the leave the plane, pick up luggage, go through immigration, customs, taxi rank… you get the drift. People like me are patient enough to walk slowly behind less mobile people and assist them, if need be). So, I had this idea, which I’d have very few better venues to share other than with you problem-solving lot. Could plane seat arrangements be made mobile and configurable prior to every flight?

While doing that for wheelies would be noble, it’d be hard to get the business case through. So, here comes the upscoping. I’m sure you quants would be aware of Bharat P Bhatta’s paper (Journal of Revenue and Pricing Management, 2013) on weight-based airfaring: http://www.smh.com.au/travel/travel-news/airfares-should-be-pay-what-you-weigh-professor-20130325-2gp12.html

And you may also be aware that Samoa Air started doing just that last month: http://edition.cnn.com/2013/04/02/travel/samoa-air-fare-by-weight

There obviously are a lot of criticisms against this idea, which TBH does sound like an economist’s armchair pipe dream. Most of them are related to discrimination, and no one wants to do that, of course. I even wondered whether companies would slant toward female Asian women for their travelling salesforce (which, according to some willing customers, that would not be unwelcome). While many people would nonetheless agree with overweight people paying more (as we know, those gluttons are sinners… not!), no one – and I repeat: no one! – would agree with charging more of wheelies. Course not. And ‘course it can be easily excluded from the weight calc. And so can any other vital equipment (no, iPads are not vital!… ok, unless adapted for medical use).

However, one must ask whether discrimination is not already in practice. It is a self-evident truth that aircraft seat space has dwindled over the last couple of decades. Is that not favouring lighter, smaller people? Under flat-rate airfare, all an airline needs to do is reduce seat space (to the point of discomfort of larger, heavier people), put more of them in the airplane, and give enough of a discount over the competition to bite into the shorties’ market (ahem, Tiger!). So, it ain’t price discrimination, but it sure is supply discrimination.

Incredibly, the biggest offender in this space is Finnair (those poor vikings!)! And we, in the beautiful Kingdom of Oz, are lucky to have thou most generous, Virgin Australia: http://www.seatguru.com/charts/shorthaul_economy.php

Libertarians amongst us (which I suspect are over-represented in this forum) will immediately say: “hey, it’s a free, competitive market; let it be”. So I won’t propose further regulation, but rather a business idea (herein thrown away into the public domain).

What if the seat space could be configurable for each passenger composition? It’s like Tetris, in’it? If you were (or are!) a large guy (no, not that, you know what I mean!) or gal (nothing wrong with that either), wouldn’t you pay more than the average airfare for a little more comfort (meaning un-numbed, fully-functional limbs that you can actually use to get off the sardine can)? Yeah, we already have some of that; it’s called business class. Or economy+ (honestly??). But, I’m talking about a fully-configurable seat plan. If you’re heavier (presumably bigger, unless extreme dense!), then yes, you’d pay more, but you’d get more than elsewhere too. Current seats are already removable, as we’d all imagine, so only the seat trains would need to be made to move slots. Yes, there’d be safety issues. And scheduling issues (to allow for reconfiguration). On the positive side, it’d mean more thorough security checks on the seats as they get moved around.

“So, what does that have to do with wheeling passengers?”, you ask. A plane like that could accommodate however many wheelies are booked in for any given flight. No bumps, no excuses.

WisdomToothBtw, Virgin Australia’s aircraft are made in Brazil, and, as we know, Brazilians just need more space: http://youtu.be/XuybWspTU1U?t=1m20s