Oh no, you mentioned the goats...

Cecil's Columns/Staff Reports
1

I spent a lot of time worrying about the goats, and you have made the same error with the plane puzzle.

The solution to the 3 door/goat problem is simplified if you simply name the goats:
NO NAME solution:
Door 1: pick prize, get shown a goat, change and get a goat
Door 2: pick goat, get shown a goat, change and get prize
D00r 3: pick goat, get shown a goat, change and get prize

therefore 2 scenarios out of 3 result in a benefit to changing.

NAME THE GOATS (e.g.Edna & Ethel)
Door 1 now has 2 possible scenarios:
pick prize, get shown Ethel, change and get Edna
pick prize, get shown Edna, change and get Ethel
Door 2, pick Ethel, get shown Edna, change and get prize
Door 3, pick Edna, get shown Ethel, change and get prize

therefore 2 scenarios out of 4 result in a benefit to changing, 2 out of 4 result in a benefit to not changing. Changing therefore has no beneficial effect.

Now that the goats are sorted, to the plane. Groundspeed and airspeed are only independent once the aircraft is moving. If you consider any speed of forward movement below take off speed, the conveyer belt moves the plane back to match it’s forward speed, so relative to the ground (& therefore the surrounding air) it remains at that speed. It can only accelerate by moving faster than the conveyor, which is not allowed by the defining parameters in the puzzle.

In a ‘real world’ scenario though, there would be a delay between the monitoring of the forward speed and the increase of the belt that would allow the speed to creep up gradually to take off speed and the plane takes off. This is not what the puzzle parameters allow, and the moment the speed is registered (e.g.1mph) it will not increase however much thrust is generated.

The intuitive way is to consider the reverse scenario - a forward moving landing airplane without reverse jet thrust will stop by wheel brakes alone, and can be held against the brakes when gunning the engine to take off.

David G.
London UK

2

The problem with this line of reasoning is that the four options are not equally likely. The two scenarios where you pick the prize only have a 17% chance of occurring (assuming Monty has a 50% chance of choosing one goat rather than the other), but the two goat options each have a chance of 33%.

Two scenarios out of four do indeed result in a benefit if you change, but those scenarios only have half the chance of occurring as the other two.

3

Tevildo has already explained why you’re wrong about the goats.

(By the way, I have simulation programs written – just for fun – in Ada, C++, Java, Perl, and Ruby, that all demonstrate positively that switching is good, simply by playing thousands of random games and counting the results. Maybe I should do a JavaScript version and put it on-line.)

You’re wrong about the airplane because, as Cecil points out, the treadmill doesn’t pull the airplane back, because the airplane is being driven by some kind of reaction engine. Assuming frictionless wheel bearings, the treadmill will have zero effect. Assuming reasonable wheel bearings, the treadmill will have a slight effect, but it won’t become meaningful unless it moves so fast that the wheel bearings melt down – in which case the treadmill will have self-destructed a lot sooner. (It’s not fair to assume that the treadmill is made of unobtainium, but the wheels aren’t.)

4

Just to keep things legal:

Monty Hall column

Treadmill column

5

I can’t say much about doors and Monty Haul (it being a little before my time), but I can say that the example of the family provided by Jordan Drachman is false.

  1. The gender of a child is not dependent on the gender of its siblings.
  2. The odds of a child being a boy or a girl are 1:1.
  3. Therefore the odds of the other child in the above family being a boy are 1/2.

Len Ragozin provides the following (flawed) description of where the 2/3 figure comes from…

We have established that order is irrelevant in the case of determining the gender of the unknown child (since gametes are all independently assorted), so the options collapse simply into boy or girl. The addition of order as a variable is unecessary; serving only to muddy the waters and produce flawed results.

6

Sorry, spacerev, this has been done to death. Your analysis is wrong. Order doesn’t matter, but what matters is that you’re told they have at least one daughter, without specifying which kid it is. The probability that the other is a boy is 2/3.

7

On second thought, that explanation is probably not too helpful. Think of the entire population of families with two kids. 50% have a boy and a girl, 25% have two girls, and 25% have two boys. Do you agree with that?

You’re told that this family could be in the either of the first two groups, but is definitely not in the third. What’s the probability that they are in the first, given this knowledge? 2/3.

8

Regarding the plane:

It helps to draw a simple force diagram. The only forces acting upon the plane are:

  1. Gravity.
  2. Normal force
  3. Friction
  4. Air resistance
  5. Thrust
    Of these, Thrust is by far the most important. Gravity and the Normal Force cancel out. As long as the thrust is sufficient to overcome the air resistance and the friction (which is why there are wheels, not skids), the plane takes off.