Chapter 7: Q8E (page 451)
Question: What is the probability that a five-card poker hand contains the ace of hearts?
Answer
The probability that a five-card poker hand contains the ace of heart is
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Question: Suppose that and are the events that an incoming mail \({E_1}\)message contains the words \({w_1}\) and \({w_2}\), respectively. Assuming that \({E_1}\) and \({E_2}\) are independent events and that \({E_1}\left| S \right.\) and \({E_2}\left| S \right.\) are independent events, where S is the event that an incoming message is spam, and that we have no prior knowledge regarding whether or not the message is spam, show that
\(p(S|{E_1} \cap {E_2}) = \frac{{p({E_1}|S)p({E_2}|S)}}{{p({E_1}|S)p({E_2}|S) + p({E_1}|\bar S)p({E_2}|\bar S)}}\)
Question: A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.9, and the probability that it is received incorrectly (as a 1) is 0.1. When a 1 is sent, the probability that it is received correctly is 0.8, and the probability that it is received incorrectly (as a 0) is 0.2
a) Find the probability that a 0 is received.
b) Use Hayes theorem to find the probability that a 0 was transmitted, given that a 0 was received.
Question: Would we reject a message as spam in Example 4
a) using just the fact that the word “undervalued” occurs in the message?
b) using just the fact that the word “stock” occurs in the message?
Suppose that a test for opium use has a 2% false positive rate and a 5% false negative rate. That is, 2% of people who do not use opium test positive for opium, and 5% of opium users test negative for opium. Furthermore, suppose that 1% of people actually use opium.
a)Find the probability that someone who tests negative for opium use does not use opium.
b) Find the probability that someone who tests positive for opium use actually uses opium.
Question: (Requires calculus) Show that if is an infinite sequence of pair wise disjoint events in a sample space S, then [ .[Hint: Use Exercise 36 and take limits.]
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