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    Bits and Set Outcomes

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    How many bits will it take to represent the following sets of outcomes?
    (For example, 1 bit is enough to represent male/female, and 2 bits to identify four seasons of spring, summer, fall, winter as of 22 = 4)

    a. The uppercase alphabet A, B, . . . , Z
    b. The digits 0, 1, . . . , 9
    c. The seconds in a 24-hour day
    d. The people in the United States (about 300,000,000 of them)

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    Solution Preview

    The minimal number of bits, n, that we need to represent a set with N members is the minimal number which satisfies:

    N =< 2^n (N smaller or equal to 2^n)

    so, taking the natural logarithm of both sides

    ln (N) =< n*ln(2)

    n >=ln(N)/ln(2) = ...

    Solution Summary

    This solution determines how many bits it will take to represent different sets of outcomes.