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Chapter 6: Problem 28
Two dice are rolled, one blue and one red. How many outcomes are doubles? (A double occurs when both dice show the same number.)
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Refer to \(a\) bag containing 20 balls-six red, six green, and eight purple. In how many ways can we select five balls if the balls are considered distinct?
Find the probability that among \(n \geq 3\) persons, at least three people have birthdays on the same month and date (but not necessarily in the same year). Assume that all months and dates are equally likely, and ignore February 29 birthdays.
Find the sum $$1 \cdot 2+2 \cdot 3+\cdots+(n-1) n$$.
Use mathematical induction to show that if \(E_{1}, E_{2}, \ldots, E_{n}\) are events, then $$ P\left(E_{1} \cup E_{2} \cup \ldots \cup E_{n}\right) \leq \sum_{i=1}^{n} P\left(E_{i}\right) $$
Show that for any events \(E_{1}\) and \(E_{2}\). $$ P\left(E_{1} \cap E_{2}\right) \geq P\left(E_{1}\right)+P\left(E_{2}\right)-1 $$
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