Aviation brain teasers

[jkstark]

Add to that the fact that there is the chance that he gets his seat, or he does not. It is simply yes or no - what the precursor to that event happening is really inconsequential. Either he gets his own seat or he does not - those are the only two choices.

[Petertenthije]

Add to that the fact that there is the chance that he gets his seat, or he does not. It is simply yes or no - what the precursor to that event happening is really inconsequential. Either he gets his own seat or he does not - those are the only two choices.

By that logic the chance of you winning the lotery are 50%. You either win or you don't. If that were really your odds I would happily buy you a ticket and split the winnings. I'd even be generous and offer you a 60/40 split.

The clincher in this riddle is that each person will take his seat, assuming it has not been taken. If it were a full free-for-all the odds would be 1/147 * 1/146 * 1/145 etc.

[jkstark]

True - so more accurately the chances of anybody getting their own seat are 50:50, and unlike the lottery, you have a choice of yes or no for each seat, as opposed to a 1 in (say) 40 chance of hitting each number out of a set of 6-8 to be right.

[Gabriel]

True - so more accurately the chances of anybody getting their own seat are 50:50, and unlike the lottery, you have a choice of yes or no for each seat.

Hum, no.
Binary mutually exclusive results have 50-50 chances only if the odds of each of them are equal, like in tossing a coin.
As a counter-example, imagine the event "hole-in-one" in Golf. You hit the ball aiming to the hole and you either put the ball in the hole or you don't. It's a clear yes-no situation, and all Glof players always aim to put the ball in the hole in the first hit (well, in par 3 holes at least). However, the rate that hole-in-one's happen is nowhere near 50%.

In this particular case, you say "the chances of anybody getting their own seat are 50:50", and that is wrong. For example, pax 2 WILL get his own seat as long as it wasn't occupied by pax 1 or, in other words, as long as pax 1 picks out of the 1476 available seat, any of the 146 seats that is not pax 2's seat. So, for the second passenger, the chances to getting his own seat are 146/147, nowhere close to 50%.

[Gabriel]

Pax 1 (1 / 147)
Pax 2 (1 / 147) * (145 / 146)
Pax 3 (1 / 147) * (145 / 146) * (144 / 145)
Pax 4 (1 / 147) * (145 / 146) * (144 / 145) * (143 / 144)
Pax 5 (1 / 147) * (145 / 146) * (144 / 145) * (143 / 144) * (142 / 143)

If you repeat this 147 times and add up the figures you end up with 0,50.


Bit more explanation.


The odds that passengers one takes your seat are 1 in 147.

The odds that passenger two does NOT take your seat are 100% if passenger one has the correct seat. However, if passenger one has the wrong seat then passenger two will pick one seat from the remaining 146 seats. That means that the odds of him NOT taking your seat are 145 to 146 IF passenger one has the wrong seat.

So the odds of passenger two NOT taking your seat are (1/147) * (145/146)

Repeat this for the amount of passengers and your done. It does not matter how many passengers there are, the end result will always add up to 50%.

Well, please help me understand.

To begin with, a small correction: you can do the above 146 times (n-1), not 147 (n). The line #146 will look like:
1/147*145/146*144*145*....*2/3*1/2.

Another way ton write the same lines:

1/147*146/146
1/147*145/146
1/147*144/146
1/147*143/146
...
1/147*2/146
1/147*1/146

In general, if I call the bottom line i=1 and the first line i=146, and n=147, the lines can be written as:
1/n * i/(n-1)

And the sum of all the lines:

Sum (from i=1 to n-1) [1/n * i/(n-1)]

And yes, the result of that is 0.5 for n=147 or whatever number. So far so good. But now the problem stars.

I don't understand your logical/statistical explanation. Yes, it gives the right result, but there are lots of wrong ways to get the right result (or see the recent exchange with jkstark). I want to make clear that I don't mean that your explanation is wrong, just that I don't understand it. And here is why:

First line: yes, 1/147 are the chances that the first pax picks the last pax's seat.

Second line:
"The odds that passenger two does NOT take your seat are 100% if passenger one has the correct seat." (I assume that "you" above is "the last passenger")
What is "correct seat"? I'd say that "The odds that passenger 2 does NOT take your seat are 100% if passenger 1 has any seat except that of pax 2", which, by the way and for whatever it's worth, will happen with a chance of 146/147.
"However, if passenger one has the wrong seat" (I assume you mean pax 2's seat) "then passenger two will pick one seat from the remaining 146 seats. That means that the odds of him NOT taking your seat are 145 to 146 IF passenger one has the wrong (pax 2's) seat."
Making that correction, ok.

"So the odds of passenger two NOT taking your seat are (1/147) * (145/146)"

Hmmm, no. That is the chances of: Pax 1 takes the seat of pax 2 AND pax 2 doesn't take your seat. However, pax 2 will also NOT take your seat if pax 1 picked a seat other than that of pax 2, which has a chance of 146/147. Since both events are mutually exclusive (because pax 1 will either pick pax 2's seat or not), then the total probability of pax 2 NOT taking your seat is the sum of the probabilities of each mutually exclusive event that yields that result, so it's 1/147*145/146 + 146/147 = (145/146 + 146) / 147. (or I am doing something wrong?)

Anyway, you say that the first line is the probability that the first passengers DOES take your seat and the second line is the probability that the second passenger DOESN'T take your seat. And then you add them (together with the rest of the lines). I don't understand why add probabilities of two events that have yield different outcomes. One normally adds the probabilities of all the mutually exclusive events that yield the SAME outcome (not OPPOSITE ones) to find the total probability for that outcome.

And I also don't understand what would be the conceptual meaning (explanation) for the third line (and following ones, but let's stay with the third one).

In short. I don't understand a thing...

[Evan]

Let's say for the sake of argument this is Spirit Airlines.

What are the chances that the last passenger will get his bag in the overhead?

(Hint: it is very unlikely)

[Gabriel]

Let's say for the sake of argument this is Spirit Airlines.

What are the chances that the last passenger will get his bag in the overhead?

(Hint: it is very unlikely)

Depends. Did he pay for his carry-on?
Since Spirit is one of the very few airlines that charge you for carry-on baggage (except one small personal item that fits under the seat), if you don't pay for it you won't get your bag in the carry on, but if you pay... it must be one of the emptiest overhead bins out there.

[brianw999]

Chances are "evens". He either does or does not sit in his assigned seat.

Soooooooooooo.....referring to my answer and your specific question "what are the chances ?...." do I assume that I gave the correct answer first ?

[Gabriel]

Soooooooooooo.....referring to my answer and your specific question "what are the chances ?...." do I assume that I gave the correct answer first ?

Yes, you were the first one to give the right answer to the specific question, as it has already been duly and timely acknowledged. But the answer to the problem (which required a correct explanation too) wasn't right.

Let me quote myself in post #52, "That's the answer to the question: The chances that last passenger sits in his assigned seat is 50%. (or "even", as Brian said before but for the wrong reason)".

And as I've said in my last post, "there are lots of wrong ways to get the right result".

Can we say that you got the right chances by chance? Or that you were right by mistake?

[3WE]

Let's say for the sake of argument this is an A-320/21.

What are the chances that the last passenger will get his bag in the overhead?

(Hint: it is very unlikely)

Fixed.

[Gabriel]

And A319 too.

[3WE]

So, if your stall speed is 60 kts and you are flying on an Easterly course at 70 KIAS in a 20 kt wind from the North and you make too rapid of a turn to the south, is there a chance of stalling from the tailwind you have picked up?

[Gabriel]

So, if your stall speed is 60 kts and you are flying on an Easterly course at 70 KIAS in a 20 kt wind from the North and you make too rapid turn to the south, is there a chance of stalling from the tailwind you have picked up?

Finally a trully aviation-related teaser!!! And a classic one!!!

By now, I'll just say that if your stall speed (1G stall speed) is 60kts and you are flying at 70kts, you are too close to stall already and there is a good chance that your too rapid (steep, accelerated) turn to downwind will make you stall.

[3WE]

Indeed, Gabriel, there are a number of dangers involved with turning downwind. Yes, they are managable dangers, although there's also a crash rate that sadly greater than zero from downwind turns!

But as a respected and wise and now departed aviation expert stated: it's important to do it gradually, giving the plane time to adjust and compensate for the tail wind...

[brianw999]

So, if your stall speed is 60 kts and you are flying on an Easterly course at 70 KIAS in a 20 kt wind from the North and you make too rapid of a turn to the south, is there a chance of stalling from the tailwind you have picked up?

If the turn is the only thing you do, no climbing, descending or power changes then .......

Answer to the question is No.

Here's a rationale for that answer. It is the summary taken from this rather interesting article. http://www.gbfs.co.za/images/Dragons.pdf ...

Summary of Dragons
A downwind turn holds no aerodynamic problems, but a climbing, low level downwind turn is a whole lot more dangerous than we imagine:
• The shallow climb reduces your ability to clear the power lines. It may also cause you to ease back and lose airspeed and lift.
• The increased groundspeed limits your ability to turn away from obstacles. It can also fool you into throttling back. or easing back and running out of airspeed and lift.
• You can get dragged down by downdraughts caused by trees, hills or buildings.
• windshear causes a loss of airspeed and lift. It can also make you overbank at a time when
you are already in trouble.
• All turns cause a loss of climb performance.
• Apparent slip can cause
you to use too much rudder then opposite aileron and up elevator – inducing a spin.
• Your reason for doing the turn may contain its own dragons.

And a slightly more tongue in cheek view of the problem by aviation writer Peter Garrison....

The following article is from the January 2005 print issue.

"Nope," I said. "No way. There's no way that a turn downwind, or upwind, or in any other direction, is any different from a turn in still air."

"Well, sonny," said the Old-Timer, "maybe out your way the air's made of different stuff. But I've buried enough pilots to know that in these parts, when that wind comes around an' hits ya from behind, if ya don' have enough of a cushion, you're just gonna stall. There's no two ways about it. lt's just common sense. That wind's gonna hit ya from behind, and that's all she wrote. Y'all writers, y'all're so used to smoke and mirrors, y'all can't see what's right in front of ya."

As he finished saying these words, an odd smirk played upon his lips.

I thought best not to insist. Thanking my companion for the favor of his company, I tossed my bag into the back seat of my car and drove off.

My first inkling of something strange came on the highway. I was doing about 70 or so, and I noticed that whenever I turned my head ever so little, I got a ditty feeling. I chalked it up to having skipped lunch. But I had to change lanes to turn into the parking lot of the Alamo Inn, and when I turned my head to look for traffic behind me, I got a painful jolt. Later, as I meditatively nursed my whiplash, I realized what must have caused it. My head had been moving forward at maybe 30 miles an hour. When I turned to look behind me, my head was suddenly going backwards at 30 miles an hour. The 60-mph change in velocity had taken place in less than a second. Small wonder I had hurt myself.

The following day the weather was terrible, and I decided that rather than fly to my next stop, which was less than 100 miles away, l'd take the train and return in the evening. My trip was uneventful, and actually quite pleasant; it was nice to be able to read and leave the driving to someone else. At least it was pleasant until I decided to visit the restroom. I got out of my seat and walked toward the back of the car. At a certain point I realized that a couple of other people were already waiting in line there, and so I turned around to try one farther forward.

I awoke to find myself supine on the floor of the railroad car, with several concerned fellow-passengers bending over me. I was breathing with difficulty, but soon recovered myself and crept to my feet. Reassuring everyone that I was all right, I regained my seat, where I resolved to stay put until the train had come to a stop in the station.

It was clear to me now what had taken place. Walking toward the back of the car, I had been moving backward at about 66 mph — assuming that the speed of the train was 70. When I turned around and took a step in the other direction, I was now moving forward at 74 mph — a change in velocity of 140 mph over the space of a couple of seconds. Naturally, the violent acceleration had hurled me backward and knocked the wind out of me. I resolved to be more aware of my surroundings in the future.

At the end of the week I was obliged to take an airline flight. By now I was all too aware of the dangers of sudden changes of direction on a moving conveyance, and was firmly resolved to spin no more. In the course of the flight I was once again obliged to heed a call of nature, and once again I started of in the wrong direction. But I was not about to repeat the mistake I had made on the train. After all, I realized that a rapid about-face now would entail a velocity change of close to 1,000 mph, which could easily reduce me to a pink film on the aft bulkhead. Having calculated in advance that if I allowed 45 seconds to turn around in the aisle I would probably be safe from injury, I reversed my direction with exquisite care and slowness, always keeping both hands on the seat backs beside me. So absorbed was I in this process, and so alert to the first hint of an unwelcome acceleration, that I scarcely noticed that some of my fellow passengers were looking at me apprehensively. I had nearly completed my 180-degree turn when I became aware of a blunt object pressed against my back and a soft voice in my ear.

"I am Air Marshal McGraw," it said. "If you'll just quietly cooperate, sir, there doesn't have to be any trouble."

I had a terrible time explaining myself, because the TSA people simply refused to grasp the simple physics of the situation, even when I carefully diagrammed the inertial reference frame for them on a sheet of graph paper, complete with arrows and subscripted V's for "velocity" — V0, V1, and so on. They seemed totally unaware of the terrible dangers attendant upon any change of direction by a pedestrian in a moving airliner. I'm surprised that accidents are not more frequent; no doubt the airlines are at pains to cover them up.