Analyzing Coin Toss Experiments with Discrete Distributions

Simulate fair and biased coin tosses to study different discrete distributions

Start with the simplest scenario: repeatedly toss a fair coin ( =

) until the first head appears.

The number of tails before the first head follows a GeometricDistribution. Simulate 100 experiments:

In[1]:=

RandomVariate[GeometricDistribution[1/2],10];ListPlot[RandomVariate[GeometricDistribution[1/2],100],FillingAxis]

Out[1]=

Compute the probability that at least five tosses are required before the first head occurs:

In[2]:=

Probability[x≥5,xGeometricDistribution[1/2]]

Out[2]=

Find the expected number of tails before a head:

In[3]:=

Mean[GeometricDistribution[1/2]]

Out[3]=

The number of tails before achieving four heads follows a NegativeBinomialDistribution:

In[4]:=

heads4=NegativeBinomialDistribution[4,1/2];

Plot the probability distribution of tail counts:

In[5]:=

DiscretePlot[PDF[heads4,k],{k,0,10}]

Out[5]=

Compute the probability of getting at least six tails before getting four heads:

In[6]:=

N[Probability[tails≥6,tailsheads4]]

Out[6]=

0.253906

Find the expected number of tails before four heads:

In[7]:=

Mean[heads4]

Out[7]=

When the number of tosses is predetermined, the count of heads follows a

BinomialDistribution

. Let's compare the distributions for 100 tosses of a fair coin, versus 100 tosses of a biased coin.

For 100 fair tosses ( =

), visualize the probability of obtaining each possible number of heads:

In[8]:=

heads[n_]:=BinomialDistribution[n,1/2];

In[9]:=

unbiasedPlot=ListLinePlot[Table[PDF[heads[100],k],{k,0,100}],PlotRangeAll,PlotStyleRGBColor[0.1,0.3,0.9],FillingAxis]

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Compute the probability of observing between 60 and 80 heads in 100 tosses:

In[10]:=

N[Probability[60≤x≤80,xheads[100]]]

Out[10]=

0.028444

Define a new binomial distribution where the probability of heads is 0.6 (the case of a biased coin):

In[11]:=

uheads[n_]:=BinomialDistribution[n,0.6];

The distribution and the corresponding probabilities have changed:

In[12]:=

biasedPlot=ListLinePlot[Table[PDF[uheads[100],k],{k,0,100}],PlotRangeAll,PlotStyleRGBColor[1,0.55,0.1],FillingAxis]

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Combine the two distributions into one plot to note their difference:

In[13]:=

Framed[Legended[Show[unbiasedPlot,biasedPlot,FrameTrue,FrameLabel{"Number of Heads","Probability"},PlotRangeAll],Placed[LineLegend[{RGBColor[0.1,0.3,0.9],RGBColor[1,0.55,0.1]},{"Unbiased Coin (p = 0.5)","Biased Coin (p = 0.6)"}],Right]],FrameMarginsMedium]

Out[13]=

Find the probability of getting between 60 and 80 heads when the coin is biased toward heads:

In[14]:=

Probability[60≤x≤80,xuheads[100]]

Out[14]=

0.543289

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Analyzing Coin Toss Experiments with Discrete Distributions

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Analyzing Coin Toss Experiments with Discrete Distributions

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