8.6E: Exercises - Independent Events

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 $P(A \cap F)$