and make 
draws with replacement.
What do the probability histograms look like as
increases? MONT 107N -- Understanding Randomness
Probability Histograms
February 5, 2010
Consider the box and make draws with replacement.
What do the probability histograms look like as increases?
Here is some Maple code that computes the binomial probabilities
and generates the corresponding probability histograms:
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| > | ![`assign`(Y, proc (n) options operator, arrow; [seq(`*`(binomial(n, i), `*`(`^`(`/`(3, 38), i), `*`(`^`(`/`(35, 38), `+`(n, `-`(i)))))), i = 0 .. n)] end proc); -1](images/PHist_4.gif){kind=small} |
| > | ![`assign`(X, proc (n) options operator, arrow; [seq(i, i = 0 .. n)] end proc); -1](images/PHist_5.gif){kind=small} |
| > | ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_6.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_7.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_8.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_9.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_10.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_11.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_12.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_13.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_14.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_15.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_16.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_17.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_18.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_19.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_20.gif){kind=small} ![`assign`(PH, proc (Xlist, Ylist) local i, start, n, pl; with(plots); `assign`(n, nops(Xlist)); `assign`(start, `+`(Xlist[1], `-`(.5))); `assign`(pl, {}); for i to `+`(n, `-`(1)) do `assign`(pl, `union...](images/PHist_21.gif){kind=small} |
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With 
draws:
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With 
draws:
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With 
draws:
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With 
draws (at this point, notice that
the probability histogram is starting to look more "normal"
in shape(!)
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Finally, with 
draws, and an approximating
normal curve in red:
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