[1eyedjack]

Seems to me that the 8 ball problem is effectively the same if you extend it to 9 balls (of which 8 are of equal weight). Psychologically it would seem a more daunting problem the more balls are in the air.

[Mbodell]

1eyedjack, on Sep 24 2007, 02:35 AM, said:

Seems to me that the 8 ball problem is effectively the same if you extend it to 9 balls (of which 8 are of equal weight).  Psychologically it would seem a more daunting problem the more balls are in the air.

The normal good problem with weights and balls, of which this version is kind of a subproblem (with 8 balls, not 9, which is probably why this version only has 8 balls), is you have 12 balls and know that 11 are identical in weight but 1 is EITHER heavier OR lighter. Now with 3 weightings of the scales of justice style scale (I.e., one side goes up or down or stays the same but you don't get a number answer) you have to determine the defective ball AND if the ball is heavier or lighter.

Many of these types of problems are fun to work on, but few make good interview questions IME (both as a candidate who nailed all of these questions when Microsoft used to ask them [but still though the a-ha questions were poor] and an interviewer who has interviewed a couple hundred technical people).

[Echognome]

Hannie, on Sep 22 2007, 12:20 PM, said:

Blofeld, on Sep 22 2007, 05:36 AM, said:

11. Probability of not observing a car in 30 minutes = 0.05. Therefore probability of not observing a car in 10 minutes has to be the cube root of 0.05. Probability of observing a car in 10 minutes is 1 minus this, or roughly 0.63.

Well done Owen.

Interesting as I came about the answer a different way, but still the same (up to whatever rounding issues we both have).

The underlying presumption here is that the arrival of cars is a homogeneous Poisson Process. Thus, we can simply use the Exponential Probability distribution to model our probabilities. To show this assumption is critical, imagine that we are observing cars at the end of a bridge between the time of 7:50am and 8:20am and that for some reason, the bridge doesn't open until 8am. In that case, the probability of observing a car in that 30 minute increment might be .95 and in the first 10 min increment it would be 0.

I'm not sure what constant default probability is referring to. Default probability is typically defined as the probability of defaulting on debt! Shrug.

Anyway, once we have independent time increments, we can use the exponential distribution.

Then, Pr(T > t) = Exp{-lambda*t}

We are given that the Pr(T < 30) = 0.95

or Pr (T > 30) = 1 - Pr(T < 30) = 0.05 (this is what Owen used when saying the probability we do NOT observe a car in 30 minutes)

Pr(T > 30) = Exp{-lambda*30} = 0.05

Solving for lambda yields it to be approx 0.1.

Then we can use Pr(T > 10) = Exp{-(0.1)*10} = 1/e or about 0.37.

Finally, Pr(T < 10) = 1 - Pr(T > 10) = 1 - 0.37 = 0.63.

I'll be interested to hear how Owen used the cube root.

Edit: Arend told me how Owen came about using the cube root. Using Pr(T>30) = Pr(T>10)*Pr(T>10)*Pr(T>10) as you have to not see the car in each 10 min increment. Nice way to solve it. Relies on the same independent time increment assumption.

Edit2: Bonus Questions - What's the probability of observing TWO cars in 10 minutes? Suppose you arrived at the observation point, how long do you expect to wait before you see the next car?

[cherdano]

What makes me think Owen gets the job and Echo doesn't?

[han]

Yes, I called Mr. Google and Owen gets the job.

I also talked with a google employer yesterday. He is well known to many here but I won't mention his name because google sees it all. He said Richard wasn't that far off with what he wrote in this thread so it isn't clear if Owen should be happy or sad.

[Echognome]

Cherdano, on Sep 24 2007, 09:17 AM, said:

What makes me think Owen gets the job and Echo doesn't?

Hannie, on Sep 24 2007, 03:21 PM, said:

Yes, I called Mr. Google and Owen gets the job.

Silly mathematicians living in the theoretical world.

In the real world, Gnome has the job. Owen didn't show his work.

[hrothgar]

If Owen is really going to work for Google in Mountain View, I wish him luck...

More importantly, I THOROUGHLY recommend a sushi joint in Cupertino called Kitsho
The aji no tataki is to die for and they have some great saki (Go with the dewazakura oka)

Make sure to eat at the sushi bar and order Omakase.

[han]

Echognome, on Sep 24 2007, 06:51 PM, said:

In the real world, Gnome has the job.

It's hard to argue with that.

But I would claim that Owen did show his work, and that he demonstrated a healthy dosis of common sense as well.

[Blofeld]

[Richard: I'm not likely to go to work at Google any time soon, but if I find myself in that part of California I'll try and remember the sushi recommendation.]

Matt: I showed working! Sheesh, how much do you want?

Now I'll take my dosis elsewhere.

[paulg]

Hannie, on Sep 25 2007, 12:21 AM, said:

Yes, I called Mr. Google and Owen gets the job.

Did you miss the release that Mr. Google had quit?