# 8.6E: 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. $$P(E)$$ $$P(F)$$ $$P(E \cap F)$$ Are $$E$$ and $$F$$ independent? Use probabilities to justify your conclusion. Find the following. $$P(F)$$ $$P(G)$$ $$P(F \cap G)$$ 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. $$P$$( both of them will pass statistics) $$P$$(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: $$P$$( Jane will make both connections) $$P$$(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. $$P(E)$$ $$P(F)$$ $$P(E \cap F)$$ Are $$E$$ and $$F$$ independent? Find the following. $$P(F)$$ $$P(G)$$ $$P(F \cap G)$$ 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 $$P(A \cap F)$$

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