Occupancy Probability Page 2

Now, let's determine the probability of getting all 3 numbers after four spins of our roulette wheel.

First, we calculate that there are 34 (or 81) results after 4 spins.
Notice that in this case, it is much more difficult to determine and to list these 81 results than it was when we were listing just 27 numbers.

 1 1 1 1 1 1 1 2 1 1 1 3 1 1 2 1 1 1 2 2 1 1 2 3 1 1 3 1 1 1 3 2 1 1 3 3 1 2 1 1 1 2 1 2 1 2 1 3 1 2 2 1 1 2 2 2 1 2 2 3 1 2 3 1 1 2 3 2 1 2 3 3 1 3 1 1 1 3 1 2 1 3 1 3 1 3 2 1 1 3 2 2 1 3 2 3 1 3 3 1 1 3 3 2 1 3 3 3 2 1 1 1 2 1 1 2 2 1 1 3 2 1 2 1 2 1 2 2 2 1 2 3 2 1 3 1 2 1 3 2 2 1 3 3 2 2 1 1 2 2 1 2 2 2 1 3 2 2 2 1 2 2 2 2 2 2 2 3 2 2 3 1 2 2 3 2 2 2 3 3 2 3 1 1 2 3 1 2 2 3 1 3 2 3 2 1 2 3 2 2 2 3 2 3 2 3 3 1 2 3 3 2 2 3 3 3 3 1 1 1 3 1 1 2 3 1 1 3 3 1 2 1 3 1 2 2 3 1 2 3 3 1 3 1 3 1 3 2 3 1 3 3 3 2 1 1 3 2 1 2 3 2 1 3 3 2 2 1 3 2 2 2 3 2 2 3 3 2 3 1 3 2 3 2 3 2 3 3 3 3 1 1 3 3 1 2 3 3 1 3 3 3 2 1 3 3 2 2 3 3 2 3 3 3 3 1 3 3 3 2 3 3 3 3

That list of 81 numbers contains all the results of 4 spins of a three-numbered roulette wheel.

Next, we must search that list to see how many of those 81 numbers contain "1 2 3 " in any order.
It turns out that there are 36 such occurrences:
 1 1 2 3 1 1 3 2 1 2 1 3 1 2 2 3 1 2 3 1 1 2 3 2 1 2 3 3 1 3 1 2 1 3 2 1 1 3 2 2 1 3 2 3 1 3 3 2 2 1 1 3 2 1 2 3 2 1 3 1 2 1 3 2 2 1 3 3 2 2 1 3 2 2 3 1 2 3 1 1 2 3 1 2 2 3 1 3 2 3 2 1 2 3 3 1 3 1 1 2 3 1 2 1 3 1 2 2 3 1 2 3 3 1 3 2 3 2 1 1 3 2 1 2 3 2 1 3 3 2 2 1 3 2 3 1 3 3 1 2 3 3 2 1

We see there are 36 ways out of 81 that we can get all 3 numbers after four spins of the wheel.

Therefore, the probability of that occurring is:

36 ÷ 81 = .44444444...

By now, you are probably thinking that there must be an easier way to do these calculations ... and there is.

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