Coin flip casino risk 2

Another way that we can look at
risk in this thought experiment is to consider the standard deviation
of all those individual bets. So again consider we have a thousand
different betting tables out there, and we have the opportunity to
allocate our bets however we like. We could put 200 tokens on one table,
5 on another, and so on. We’re looking at the two extreme cases
here of course, but in any case, what is the standard deviation of the
results of all those individual bet?. So we’re going to look at standard
deviation after the fact, in other words,
we’re going to assume we bet already and we’re looking at the outcome. So if we have one token per table we
might have a loss of one token on one table, a gain on another,
another loss, and a gain, and a gain and so on,
all the way across our 1,000 tables. Now, we don’t actually have to spend
much time doing the math here. It turns out that
the standard deviation for this case is pretty easy to calculate. Because for each table,
the result is either -1 or 1. And it turns out that the standard
deviation in this case is easily calculated as 1.0. So it doesn’t matter what
the particular outcome turns out to be. We know we’re either going to lose 1 or
gain 1 at every single table. And so our expected standard
deviation there is 1. The outcome for the second scenario where we make
one bet with a 1000 tokens and then essentially 0 bets on the other tables,
turns out a little bit differently. So on this first table
where we bet 1000 tokens, we either win 1000 or we lose 1000. And on all the other tables,
the outcome is exactly 0. Same for this case where
we lost on that first one. So again,
we bet on one table 1,000 tokens, and we bet 0 tokens on the other 999. Reasoning it out this way enables us
to calculate the standard deviation. Whichever way it goes whether we win or
lose, the standard deviation on
this event is the same. And the answer is 31.62. Key point here is, the standard
deviation or risk is much, much larger. Well as you can see
about 31 time larger, when we make that single bet on one
table and no bets on the other tables.

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