PROBLEM SET: INDEPENDENT EVENTS
The distribution of the number of fiction and non-fiction books checked out at a city's main library and at a smaller branch on a given day is as follows.
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MAIN (M)
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BRANCH (B)
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TOTAL
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FICTION (F)
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300
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100
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400
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NON-FICTION (N)
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150
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50
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200
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TOTALS
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450
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150
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600
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Use this table to determine the following probabilities:
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\(P(F)\)
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\(P(M | F)\)
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\(P(N | B)\)
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4. Is the fact that a person checks out a fiction book independent of the main library? Use probabilities to justify your conclusion.
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For a two-child family, let the events \(E\), \(F\), and \(G\) be as follows.
\(E\): The family has at least one boy
\(F\): The family has children of both sexes
\(G\): The family's first born is a boy
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Find the following.
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\(P(E)\)
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\(P(F)\)
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\(P(E \cap F)\)
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Are \(E\) and \(F\) independent? Use probabilities to justify your conclusion.
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Find the following.
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\(P(F)\)
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\(P(G)\)
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\(P(F \cap G)\)
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Are \(F\) and \(G\) independent? Use probabilities to justify your conclusion.
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Do the following problems involving independence.
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If \(P(E) = .6\), \(P(F) = .2\), and \(E\) and \(F\) are independent, find \(P\)(\(E\) and \(F\)).
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If \(P(E) = .6\), \(P(F) = .2\), and \(E\) and \(F\) are independent, find \(P\)(\(E\) or \(F\)).
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If \(P(E) = .9\), \(P(F | E) = .36\), and \(E\) and \(F\) are independent, find \(P(F)\).
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If \(P(E) = .6\), \(P\)(\(E\) or \(F\)) = .8, and \(E\) and \(F\) are independent, find \(P(F)\).
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In a survey of 100 people, 40 were casual drinkers, and 60 did not drink. Of the ones who drank, 6 had minor headaches. Of the non-drinkers, 9 had minor headaches. Are the events "drinkers" and "had headaches" independent?
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It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
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John's probability of passing statistics is 40%, and Linda's probability of passing the same course is 70%. If the two events are independent, find the following probabilities.
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\(P\)( both of them will pass statistics)
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\(P\)(at least one of them will pass statistics)
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Jane is flying home for the Christmas holidays. She has to change planes twice. There is an 80% chance that she will make the first connection, and a 90% chance that she will make the second connection. If the two events are independent, find the probabilities:
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\(P\)( Jane will make both connections)
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\(P\)(Jane will make at least one connection)
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For a three-child family, let the events \(E\), \(F\), and \(G\) be as follows.
\(E\): The family has at least one boy
\(F\): The family has children of both sexes
\(G\): The family's first born is a boy
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Find the following.
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\(P(E)\)
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\(P(F)\)
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\(P(E \cap F)\)
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Are \(E\) and \(F\) independent?
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Find the following.
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\(P(F)\)
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\(P(G)\)
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\(P(F \cap G)\)
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Are \(F\) and \(G\) independent?
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\(P(K|D) = 0.7\), \(P(D) = 0.25\) and \(P(K)=0.7\)
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Are events \(K\) and \(D\) independent? Use probabilities to justify your conclusion.
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Find \(P(K \cap D)\)
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\(P(R|S) = 0.4\), \(P(S) = 0.2\) and \(P(R)=0.3\)
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Are events \(R\) and \(S\) independent? Use probabilities to justify your conclusion.
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Find \(P(R \cap S)\)
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At a college:
54% of students are female
25% of students are majoring in engineering.
15% of female students are majoring in engineering.
Event \(E\) = student is majoring in engineering
Event \(F\) = student is female
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Are events \(E\) and \(F\) independent? Use probabilities to justify your conclusion.
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Find \(P(E \cap F)\)
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At a college:
54% of all students are female
60% of all students receive financial aid.
60% of female students receive financial aid.
Event \(A\) = student receives financial aid
Event \(F\) = student is female
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Are events \(A\) and \(F\) independent? Use probabilities to justify your conclusion.
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Find \(P(A \cap F)\)
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