6.6E: Exercises - Independent Events

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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.

	MAIN (M)	BRANCH (B)	TOTAL
FICTION (F)	300	100	400
NON-FICTION (N)	150	50	200
TOTALS	450	150	600

Use this table to determine the following probabilities:

\(P(F)\)	\(P(M \| F)\)
\(P(N \| B)\)	4. Is the fact that a person checks out a fiction book independent of the main library? Use probabilities to justify your conclusion.

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

Find the following.
1. \(P(E)\)
2. \(P(F)\)
3. \(P(E \cap F)\)
4. Are \(E\) and \(F\) independent? Use probabilities to justify your conclusion.

Find the following.
1. \(P(F)\)
2. \(P(G)\)
3. \(P(F \cap G)\)
4. Are \(F\) and \(G\) independent? Use probabilities to justify your conclusion.

Do the following problems involving independence.

If \(P(E) = .6\), \(P(F) = .2\), and \(E\) and \(F\) are independent, find \(P\)(\(E\) and \(F\)).	If \(P(E) = .6\), \(P(F) = .2\), and \(E\) and \(F\) are independent, find \(P\)(\(E\) or \(F\)).
If \(P(E) = .9\), \(P(F \| E) = .36\), and \(E\) and \(F\) are independent, find \(P(F)\).	If \(P(E) = .6\), \(P\)(\(E\) or \(F\)) = .8, and \(E\) and \(F\) are independent, find \(P(F)\).
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?	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?

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.
1. \(P\)( both of them will pass statistics)
2. \(P\)(at least one of them will pass statistics)

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:
1. \(P\)( Jane will make both connections)
2. \(P\)(Jane will make at least one connection)

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

Find the following.
1. \(P(E)\)
2. \(P(F)\)
3. \(P(E \cap F)\)
4. Are \(E\) and \(F\) independent?

Find the following.
1. \(P(F)\)
2. \(P(G)\)
3. \(P(F \cap G)\)
4. Are \(F\) and \(G\) independent?

\(P(K\|D) = 0.7\), \(P(D) = 0.25\) and \(P(K)=0.7\) Are events \(K\) and \(D\) independent? Use probabilities to justify your conclusion. Find \(P(K \cap D)\)	\(P(R\|S) = 0.4\), \(P(S) = 0.2\) and \(P(R)=0.3\) Are events \(R\) and \(S\) independent? Use probabilities to justify your conclusion. Find \(P(R \cap S)\)
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 Are events \(E\) and \(F\) independent? Use probabilities to justify your conclusion. Find \(P(E \cap F)\)	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 Are events \(A\) and \(F\) independent? Use probabilities to justify your conclusion. Find \(P(A \cap F)\)