Hi Jeff Leader; Re the rate of getting 38 spins in a row, assuming 50% chance for each spin...
There are a number of problems with the analysis. First, you need to take into account the fact that you can get 38 in a row in two different ways, a streak of red, and a streak of black. This doubles the calculated rate.
Second, you need to take into account the fact that probabilities are not independent. That is, if at spin number 23,456 you have a streak start with a length of 34, then it is not possible for you to have a streak of length 4 start at spin number 23,461. (I.e. spin #23,461 was in the middle of another streak, and so can't be the start of a streak.)
Third, we haven't specified whether streaks of length greater than 38 will be counted. From the way this has been calculated, I would surmise that it was not the intention to include longer streaks.
Having said all this, I suppose I have put myself in the position of actually having to make the calculation. The best way of analyzing this is to analyze it from the point of view of streaks instead of spins. That way the probabilities are independent. There is no interference between two consecutive streaks. A streak simply stops when it does, and the next streak starts afresh. But our rates are to be calculated at the spin rate, rather than at the streak formation rate.
So first of all, we have to calculate, (average) streak rate. This is the inverse of how long the average streak is.
It is pretty clear that 1/2 of all streaks are of length 1, 1/4 are of length 2, 1/8 are of length 3, etc. I will have to leave this to intuition, it is obvious to me. (But note that it is easy to make the wrong calculation. If you tried to use probabilities to calculate the average streak rate directly, rather than computing the average streak length, you could easily end up with a wrong answer. The reason is that the average rate is not the average of the rates, as longer streaks predominate. If you took this fact into account, then you would get an equivalent answer to that obtained here, the inverse of the length.)
Computing the average streak length by multiplying streak lengths by probabilities of that length, we get:
AveStreakLength = 1*1/2 + 2*1/4 + 3*1/8 + 4*1/16 + ...
Factor out 1/2 gives:
AveStreakLength = 1/2 (1 + 2*1/2 + 3*1/4 + 4*1/8 + ...)
Noting that 1/(1-x)^2 = 1 + 2x + 3x^2 + 4x^3 + ..., and substituting x=1/2, the above sum is easy to calculate, and you get:
AveStreakLength = 1/2 ( 1/(1-1/2)^2) = 2.
Now, if 2 spins are performed per minute, then the above calculation gives that there is (on average) one streak started per minute. Streaks of length 38 happen (among all streaks) at a chance of 1/2^38, just as streaks of length 1 happen with a chance of 1/2^1 = 50%.
Thus the rate at which 38-long streaks occur will be at about 1 every 2^39 spins, or one every 1.5 million years. Adding in streaks of all lengths longer than 38 brings the rate to the stated rate of one per 740,000 years.
-- Carl |
