91Ó°ÊÓ

1. A cafeteria offers a three-course meal consisting of an entree, a starch, and a dessert. The possible choices are given in the following table:

A person is to choose one course from each category.

$\left(a\right)$How many outcomes are in the sample space?

$\left(b\right)$Let $A$be the event that ice cream is chosen. How many outcomes are in$A?$

$\left(c\right)$Let $B$be the event that chicken is chosen. How many outcomes are in$B?$

$\left(d\right)$List all the outcomes in the event$AB.$

$\left(e\right)$Let$C$be the event that rice is chosen. How many outcomes are in$C?$

$\left(f\right)$List all the outcomes in the event$ABC.$

Prove the following relations:

$EF⊂E⊂E∪F$

A box contains $3$ marbles: $1$ red, $1$ green, and $1$ blue. Consider an experiment that consists of taking $1$ marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty percent wear a ring and $30$ percent wear a necklace. If one of the students is chosen randomly, what is the probability that this student is wearing

(a) a ring or a necklace?

(b) a ring and a necklace?

A $5$-card hand is dealt from a well-shuffled deck of $52$playing cards. What is the probability that the hand contains at least one card from each of the four suits?

If $P\left(E\right)=.9$and$P\left(F\right)=.8$, show that$P\left(EF\right)â‰\yen .7$.In general, prove Bonferroni’s inequality, namely$P\left(EF\right)â‰\yen P\left(E\right)+P\left(F\right)−1$.

A total of $28$ percent of American males smoke cigarettes, $7$ percent smoke cigars, and $5$ percent smoke both cigars and cigarettes.

(a)What percentage of males smokes neither cigars nor cigarettes?

(b)What percentage smokes cigars but not cigarettes?

A basketball team consists of 6 frontcourt and 4 backcourt players. If players are divided into roommates at random,what is the probability that there will be exactly two roommate pairs made up of backcourt and a frontcourt player?

An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The

classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students who are in both Spanish and French, 4 who are in both Spanish and German, and 6 who are in both French and German. In addition, there are 2 students taking all 3 classes.

(a) If a student is chosen randomly, what is the probability that he or she is not in any of the language classes?

(b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class?

(c) If 2 students are chosen randomly, what is the probability that at least 1 is taking a language class?

A basketball team consists of $6$frontcourt and$4$ backcourt players. If players are divided into roommates at random, what is the probability that there will be exactly two roommate pairs made up of a backcourt and a frontcourt player?