Over a long period (two months) two out of three of the cars stopping
at our gas station are foreign cars.  What is the probability that
three cars in a row will be American cars?  Four in a row?  Five in a
row?  Six in a row?  Seven in a row?

wy...

Assuming your 2 out of 3 value is correct, then there's
a 1 in 3 chance that any car that pulls in will be American.

This same probability holds true for each successive vehicle,
so the chances that 2 in a row will be American are 1/3 x 1/3,
or 1/9, or 1 chance in 9.

So:

3 in a row = 1/3 x 1/3 x 1/3 = 1/27
4 in a row = 1/3 x 1/3 x 1/3 x 1/3 = 1/81
5 in a row = 1/3 x 1/3 x 1/3 x 1/3 x 1/3 = 1/243
6 in a row = 1/3 x 1/3 x 1/3 x 1/3 x 1/3 x 1/3 = 1/729
7 in a row = 1/3 x 1/3 x 1/3 x 1/3 x 1/3 x 1/3 x 1/3 = 1/2187

Pretty good odds, if you're betting with your co-workers against
it happening...  ; )

A similar discussion of how this affects the probability of 
coin tosses can be found on this page from the University of
New England's School of Psychology:

"I said that six heads in a row is twice as unlikely as five.
 The derivation of this depends on the multiplication rule of
 probability for independent events. The probability of heads
 is .5, and using this rule, we find that the probability of
 two heads in a row is .5 x .5 or .25. As a rule, the
 probability of n heads in a row is .5^n. Thus, p^(5 heads in a
 row) is .5^5 or .03125 . Likewise, p^(6 heads in a row) is .56
 or .015625 . You will find that p^(5H) / p^(6H) = .03125/.015625
 = 2."
http://www.une.edu.au/WebStat/unit_materials/c5_inferential_statistics/probability.html

The chances of a coin coming up heads on each toss is 1 in 2,
or 1/2, or .5. For the chances of a car being American, I used
your long-term figure of 1 in 3, or 1/3, or .3333.

I used fractions, instead of decimals, since it's easier to 
multiply 1/3 instead of .3333, but the results are the same.

As noted on the page above, don't fall for the "gambler's
fallacy" and believe that just because the last 5 cars were
American, the chances of the very next car being American
is any different than the 1 in 3 that was true of the first
and other cars. That's simply not the case. Yet the chances
that 6 in a row will be American is still 1 in 729.


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