help coin_toss
  coin toss experiment
 
  [results, ave_value] = coin_toss (ntrials);
 
  1. rand returns a number from a pseudo-random sequence 
     based on a uniform distribution
  2. coin toss (two outcomes) simulation: let H be the
     event that rand is in (0, 0.5), let T be the event that
     rand is in (0.5, 1.0)
  3. represent H by 1 and T by 0

[results, ave_value] = coin_toss (10);
ave_value(end)

ans =

    0.4000

[results, ave_value] = coin_toss (100);
ave_value(end)

ans =

    0.4400

[results, ave_value] = coin_toss (100);
ave_value(end)

ans =

    0.4400

[results, ave_value] = coin_toss (100);
ave_value(end)

ans =

    0.4700

[results, ave_value] = coin_toss (200);
ave_value(end)

ans =

    0.5100

[results, ave_value] = coin_toss (200);
ave_value(end)

ans =

    0.4950

[results, ave_value] = coin_toss (200);
ave_value(end)

ans =

    0.5450

[results, ave_value] = coin_toss (200);
[results, ave_value] = coin_toss (200);
[results, ave_value] = coin_toss (200);
[results, ave_value] = coin_toss (200);
ave_value(end)

ans =

    0.4300

[results, ave_value] = coin_toss (1000);
ave_value(end)

ans =

    0.5140

[results, ave_value] = coin_toss (1000);
ave_value(end)

ans =

    0.5030

[results, ave_value] = coin_toss (1000);
ave_value(end)

ans =

    0.5000

[results, ave_value] = coin_toss (1000);
ave_value(end)

ans =

    0.4630

s = RandStream.create('mt19937ar','seed',1);
help RandStream
 <strong>RandStream</strong> Random number stream.
    (Pseudo)random numbers in MATLAB come from one or more random number
    streams.  The simplest way to generate arrays of random numbers is to use
    RAND, RANDI, or RANDN.  These functions all draw values from the same
    stream of uniform pseudorandom numbers, known as the global random number
    stream.  You can create other streams that act separately from the current
    global stream, and you can use their RAND, RANDI, or RANDN methods to
    generate arrays of random numbers.  You can also designate any random
    number stream you create as the global stream.
 
    To create a single random number stream, use the <strong>RandStream</strong> constructor.
    To create multiple independent random number streams, use <strong>RandStream</strong>.CREATE.
    The RNG function provides a simple interface to create a new global stream.
 
    STREAM = <strong>RandStream</strong>.GETGLOBALSTREAM returns the current global random
    number stream, i.e., the one that the RAND, RANDI, and RANDN functions
    currently draw values from.
 
    PREVSTREAM = <strong>RandStream</strong>.SETGLOBALSTREAM(STREAM) sets STREAM as the current
    global random number stream, i.e., designates it as the stream that the
    RAND, RANDI, and RANDN functions will draw values from.  PREVSTREAM is the
    stream that was previously designated as the global random number stream.
 
    A random number stream S has properties that control its behavior.  Access or
    assign to a property using P = S.Property or S.Property = P.
 
    <strong>RandStream</strong> properties:
       <a href="matlab:help RandStream/Type">Type</a>            - (Read-only) identifies the type of generator algorithm
                         used by the stream.
       <a href="matlab:help RandStream/Seed">Seed</a>            - (Read-only) the seed value used to create the stream.
       <a href="matlab:help RandStream/NumStreams">NumStreams</a>      - (Read-only) the number of streams created at the same
                         time as the stream.
       <a href="matlab:help RandStream/StreamIndex">StreamIndex</a>     - (Read-only) the stream's index among the group of streams
                         in which it was created.
       <a href="matlab:help RandStream/State">State</a>           - the internal state of the generator.  You should not depend
                         on the format of this property, or attempt to improvise a
                         property value.  The value you assign to S.State must be a
                         value read from S.State previously.  Use RESET to return a
                         stream to a predictable state without having previously read
                         from the State property.
       <a href="matlab:help RandStream/Substream">Substream</a>       - the index of the substream to which the stream is
                         currently set.  The default is 1.  Multiple substreams are
                         not supported by all generator types.
       <a href="matlab:help RandStream/NormalTransform">NormalTransform</a> - the transformation algorithm that RANDN(S, ...) uses
                         to generate normal pseudorandom values from uniform
                         values.  The property value is one of 'Ziggurat' (the
                         default), 'Polar', or 'Inversion'.
       <a href="matlab:help RandStream/Antithetic">Antithetic</a>      - a logical value indicating whether S generates antithetic
                         uniform pseudorandom values, that is, the usual values
                         subtracted from 1.  The default is false.
       <a href="matlab:help RandStream/FullPrecision">FullPrecision</a>   - a logical value indicating whether S generates values
                         using its full precision.  Some generators are able to
                         create pseudorandom values faster, but with fewer random
                         bits, if FullPrecision is false.  The default is true.
   
    The sequence of pseudorandom numbers produced by a random number stream S is
    determined by the internal state of its random number generator.  Saving and
    restoring the generator's internal state via the 'State' property allows you
    to reproduce output.
 
    Examples:
 
       Create a single stream and designate it as the current global stream:
          s = <strong>RandStream</strong>('mt19937ar','Seed',1)
          <strong>RandStream</strong>.setGlobalStream(s);
 
       Create three independent streams:
          [s1,s2,s3] = <strong>RandStream</strong>.create('mrg32k3a','NumStreams',3);
          r1 = rand(s1,100000,1); r2 = rand(s2,100000,1); r3 = rand(s3,100000,1);
          corrcoef([r1,r2,r3])
 
       Create only one stream from a set of three independent streams, and
       designate it as the current global stream:
          s2 = <strong>RandStream</strong>.create('mrg32k3a','NumStreams',3,'StreamIndices',2)
          <strong>RandStream</strong>.setGlobalStream(s2);
 
       Reset the global random number stream that underlies RAND, RANDI, and
       RANDN back to its beginning, to reproduce previous results:
          stream = <strong>RandStream</strong>.getGlobalStream;
          reset(stream);
 
       Save and restore the current global stream's state to reproduce the
       output of RAND:
          stream = <strong>RandStream</strong>.getGlobalStream;
          savedState = stream.State;
          u1 = rand(1,5)
          stream.State = savedState;
          u2 = rand(1,5) % u2 contains exactly the same values as u1
 
       Reset the global random number stream to its "factory default" initial
       settings.  This causes RAND, RANDI, and RANDN to start over, as if in a
       new MATLAB session.
          s = <strong>RandStream</strong>('mt19937ar','Seed',0)
          <strong>RandStream</strong>.setGlobalStream(s);
 
       Reinitialize the global random number stream using a seed based on the
       current time.  This causes RAND, RANDI, and RANDN to return different
       values in different MATLAB sessions.  NOTE: It is usually not desirable
       to do this more than once per MATLAB session.
          s = <strong>RandStream</strong>('mt19937ar','Seed','shuffle')
          <strong>RandStream</strong>.setGlobalStream(s);
 
       Change the transformation algorithm that RANDN uses to create normal
       pseudorandom values from uniform values.  Note that this does not
       replace or reset the global stream.
          stream = <strong>RandStream</strong>.getGlobalStream;
          stream.NormalTransform = 'inversion'
 
    <strong>RandStream</strong> methods:
        <a href="matlab:help RandStream/RandStream">RandStream/RandStream</a>  - Create a random number stream.
        <a href="matlab:help RandStream.create">create</a>                 - Create multiple independent random number streams.
        <a href="matlab:help RandStream.list">list</a>                   - List available random number generator algorithms.
        <a href="matlab:help RandStream.getGlobalStream">getGlobalStream</a>        - Get the current global random number stream.
        <a href="matlab:help RandStream.setGlobalStream">setGlobalStream</a>        - Replace the global random number stream.
        <a href="matlab:help RandStream/reset">reset</a>                  - Reset a stream to its initial internal state.
        <a href="matlab:help RandStream/rand">rand</a>                   - Pseudorandom numbers from a uniform distribution.
        <a href="matlab:help RandStream/randn">randn</a>                  - Pseudorandom numbers from a standard normal distribution.
        <a href="matlab:help RandStream/randi">randi</a>                  - Pseudorandom integers from a uniform discrete distribution.
        <a href="matlab:help RandStream/randperm">randperm</a>               - Random permutation.
 
    See also <a href="matlab:help rng">rng</a>, <a href="matlab:help randfun/rand">randfun/rand</a>, <a href="matlab:help randfun/randn">randfun/randn</a>, <a href="matlab:help randfun/randi">randfun/randi</a>.

    Reference page in Help browser
       <a href="matlab:doc RandStream">doc RandStream</a>

RandStream.setGlobalStream(s);
rand

ans =

    0.4170

rand

ans =

    0.7203

s = RandStream.create('mt19937ar','seed',1);
RandStream.setGlobalStream(s);
rand

ans =

    0.4170

rand

ans =

    0.7203

[results, ave_value, rel_unc] = coin_toss (10);
rel_unc(end)

ans =

    0.3969

[results, ave_value, rel_unc] = coin_toss (100);
rel_unc(end)

ans =

    0.4898

% I had an error in my code, which I fixed; corrected results appear below

[results, ave_value, rel_unc] = coin_toss (10);
rel_unc(end)

ans =

    0.3691

[results, ave_value, rel_unc] = coin_toss (100);
rel_unc(end)

ans =

    0.1044

[results, ave_value, rel_unc] = coin_toss (200);
rel_unc(end)

ans =

    0.0789

[results, ave_value, rel_unc] = coin_toss (1000);
rel_unc(end)

ans =

    0.0315

[results, ave_value, rel_unc] = coin_toss (2000);
rel_unc(end)

ans =

    0.0226

[results, ave_value, rel_unc] = coin_toss (2000);
rel_unc(end)

ans =

    0.0223

figure(2)
plot(rel_unc,'k*-')