Lecture 23: Binomial Theorem (continued), Principles of Counting

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Problem of the day

If a Cardinal can pray a soul out of purgatory, by himself, in 1 hour, a Bishop in 3 hours, a Priest in 5 hours, and a Friar in 7 hours, how long would it take them to pray 4 souls out of purgatory, all praying together?

Assignment

Section 8.4, problems 2n+1, n=0,...,23; and 55-57.
Section 8.5, problems 2n+1, n=0,...,25
Section 8.6, problems 2n+1, n=0,...,22

Brief review of the lecture

  • Some examples and uses of the Binomial Theorem were discussed. For example, a problem similar to today's problem of the day was solved.

  • Two principles of Counting were introduced, they are

    The Multiplication Principle

    If E1 and E2 are two events, and E1 can occur in m ways and once E1 has occured, E2 can occur in n ways, then E1, E2 can both occur in (m x n) ways.

    The Addition Principle

    If E1 and E2 are disjoint events (i.e. mutually exclusive) and E1 happens in m ways and E2 happens in n ways, then E1 or E2 can happen in (m + n) ways.

  • Populations and Samples. To successfully answer questions concerning counting and probability, you must first be able to answer the following two questions :

    1. Does the experiment allow replacement ?
    2. Does the order of the outcomes matter ?

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