1. ## A dice puzzle

Hi,

I'm not a university student (I was, but that was many years ago). Nonetheless, I'm trying to solve a probability puzzle, and this seems like the best place on the Internet to ask for help. The puzzle is as follows:

--------------------------------

N people roll two dice.

After the first roll, half of the people are randomly eliminated, and the
other half get to roll both dice a second time. This continues until all
the people are eliminated. (Thus, N/2 people roll the dice exactly once;
N/4 people roll them exactly twice, N/8 people roll them exactly three
times, etc).

When the process is complete, how many of the N people will have rolled
double sixes at least twice? How many will have rolled double sixes at
least three times?

---------------------------------

Any insights you have would be greatly appreciated. I took probability courses years ago, but I've forgotten a lot of it, and I don't even think that problems like this are anything I ever learned to solve!

Thanks very much.

Hi,

I'm not a university student (I was, but that was many years ago). Nonetheless, I'm trying to solve a probability puzzle, and this seems like the best place on the Internet to ask for help. The puzzle is as follows:

--------------------------------

N people roll two dice.

After the first roll, half of the people are randomly eliminated, and the
other half get to roll both dice a second time. This continues until all
the people are eliminated. (Thus, N/2 people roll the dice exactly once;
N/4 people roll them exactly twice, N/8 people roll them exactly three
times, etc).

When the process is complete, how many of the N people will have rolled
double sixes at least twice? How many will have rolled double sixes at
least three times?

---------------------------------

Any insights you have would be greatly appreciated. I took probability courses years ago, but I've forgotten a lot of it, and I don't even think that problems like this are anything I ever learned to solve!

Thanks very much.

A couple of suggestions:

1. N needs to be a power of two for this to work: $N = 2^m$

2. Start with a concrete value of N to get the feel of things. Perhaps N = 8 (or even N = 4).

3. First consider at least once = 1 - Pr(no-one). Then at least twice = 1 - Pr(no-one or one).

Hi,

I'm not a university student (I was, but that was many years ago). Nonetheless, I'm trying to solve a probability puzzle, and this seems like the best place on the Internet to ask for help. The puzzle is as follows:

--------------------------------

N people roll two dice.

After the first roll, half of the people are randomly eliminated, and the
other half get to roll both dice a second time. This continues until all
the people are eliminated. (Thus, N/2 people roll the dice exactly once;
N/4 people roll them exactly twice, N/8 people roll them exactly three
times, etc).

When the process is complete, how many of the N people will have rolled
double sixes at least twice? How many will have rolled double sixes at
least three times?

---------------------------------

Any insights you have would be greatly appreciated. I took probability courses years ago, but I've forgotten a lot of it, and I don't even think that problems like this are anything I ever learned to solve!

Thanks very much.

As Mr Fantastic observes $N$ must be a power of $2$, then the number of times the die are rolled is:

$K=N+N/2+N/2^2+ ...+1=1+2+2^2+...+2^{\log_2(N)}$

The right most expression is a finite geometric series and so:

$
K=\frac{1-2^{1+\log_2{N}}}{1-2}
$

RonL

4. Originally Posted by CaptainBlack
As Mr Fantastic observes $N$ must be a power of $2$, then the number of times the die are rolled is:

$K=N+N/2+N/2^2+ ...+1=1+2+2^2+...+2^{\log_2(N)}$

The right most expression is a finite geometric series and so:

$
K=\frac{1-2^{1+\log_2{N}}}{1-2}
$

RonL
I'm probably misunderstanding your reply CaptainB, but the question asks how many (expected number?) of the N people will have rolled double sixes at least twice etc., not how many (expected number?) of the total number of rolls are double sixes etc. (which is how I originally analysed it .....)