Suppose you flip fair coins.
(a) How many possible outcomes (microstates) are there?
(b) What is the probability of getting the sequence HTHHTTTHTHHHTHHHHTHT (in exactly that order)?
(c) What is the probability of getting heads and tails (in any order)?
(b) The probability of getting the sequence is
(c)The probability of getting 12 heads and 8 tails are
(a)We suppose we flip 20 coins, . The total number of microstates will be:
So, the answer is or 1048576 microstates.
(b)The likelihood of obtaining any specific sequence of heads and tails (for example HTHHTTTHTHHHTHHHHTHT) is
As a result, the answer is for the sequence of heads and tails. (for example HTHHTTTHTHHHTHHHHTHT) is or .
(c)Regardless of order, the probability of getting the macrostate of 12 heads and 8 tails is given by:
where n denotes the number of heads and N denotes the number of coins, so:
After arranging we get,
Finally solution is
The natural logarithm function, , is defined so that for any positive number .Sketch a graph of the natural logarithm function. Prove the identitieslocalid="1650331641178" and localid="1650331643409" (c) Prove that localid="1650331645612" . (d) Derive the useful approximation
which is valid when localid="1650331651790" . Use a calculator to check the accuracy of this approximation for localid="1650331654235" and localid="1650331656447"
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