Chapter 7: Q21E (page 467)
To determine the smallest number of people you need to choose at random so that the probability that at least two of them were both born on April exceeds.
Answer
The smallest number of people is .
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Question: Prove Theorem \(2\), the extended form of Bayes’ theorem. That is, suppose that \(E\) is an event from a sample space \(S\) and that \({F_1},{F_2},...,{F_n}\) are mutually exclusive events such that \(\bigcup\nolimits_{i = 1}^n {{F_i} = S} \). Assume that \(p\left( E \right) \ne 0\) and \(p\left( {{F_i}} \right) \ne 0\) for \(i = 1,2,...,n\). Show that
\(p\left( {{F_j}\left| E \right.} \right) = \frac{{p\left( {E\left| {{F_j}} \right.} \right)p\left( {{F_j}} \right)}}{{\sum\nolimits_{i = 1}^n {p\left( {E\left| {{F_i}} \right.} \right)p\left( {{F_i}} \right)} }}\)
(Hint: use the fact that \(E = \bigcup\nolimits_{i = 1}^n {\left( {E \cap {F_i}} \right)} \).)
Question 23. To determine
What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up heads?
Question: Suppose that andare events in a sample space andand . Find.
Question: Suppose that a Bayesian spam filter is trained on a set of 1000 spam messages and 400 messages that are not spam. The word “opportunity” appears in 175 spam messages and 2 messages that are not spam. Would an incoming message be rejected as spam if it contains the word “opportunity” and the threshold for rejecting a message is 0.9?
Question: What is the probability that a five-card poker hand contains cards of five different kinds.
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