Probabilities and Statistics: Calculating Probabilities and Venn Diagrams

Exercises 2.2

1) What does it mean for two events to be independent?

2) Suppose you are dealt one card each from two separate decks of cards. What is the probability that both of your cards are:

a) red?

b) spades?

c) jacks?

d) face cards?

3) For each situation below, determine whether the two events are independent.

a) Flip a coin and then draw a card from a standard deck of 52 cards.

b) Draw a marble from a bag, do not replace it, and then draw a 2nd marble from the same bag.

4) A spinner with three equal spaces of red, blue, and green is spun one time. A single six-sided die is rolled once. What is the probability that you get blue and a number greater than 3?

5) Suppose you are dealt two cards, one after another from a standard deck of cards. What is the probability that both of your cards are:

a) spades?

b) the same suit?

c) kings?

8) A basket contains 8 female guinea pigs. Suppose that 5 of the 8 female guinea pigs are pregnant. Three guinea pigs are selected from the basket at random with replacement. What is the probability that:

a) all three selected guinea pigs are pregnant?

b) none of the three selected guinea pigs are pregnant?

9) A classroom contains 12 males and 18 females. Two students will be randomly selected to give speeches. What is the probability that the two students who give speeches are:

a) two females?

b) two males?

c) 1 male and 1 female (in either order)?

10) If 18% of all Americans are underweight, find the probability two randomly selected Americans will both be underweight.

11) A survey found that 68% of book buyers are 40 years old or older. If two book buyers are selected at random, what is the probability that both are 40 years old or older?

12) The Gallup Poll reported that 82% of Americans used a seat belt the last time they got into a car. If four people are selected at random, find the probability that they all used a seat belt the last time they got into a car.

15) At a local university, 70% of all incoming freshmen have computers. If three students are selected at random, what is the probability that:

a) none have computers?

b) all three have computers?

Exercises 2.3

4) Consider each event. Decide whether each pair of outcomes are mutually exclusive.

a) Roll a die: Get and even number and get a number less than 3.

b) Roll a die: Get a prime number (2, 3, or 5) and get a six.

c) Roll a die: Get a number greater than 3 and get a number less than 3.

d) Select a student: Get a student with blue eyes and get a student with blond hair.

e) Select a college student: Get a sophomore and get a student that is a math major.

f) Select a course: Get an Algebra course and get an English course.

g) Select a voter: Get a Republican and get a Democrat.

5) There are 200 male students at a particular school. Of these, 58 play football, 40 play basketball, and 8 play both.

a) Draw and label a Venn diagram for this situation. (Be sure to label all 4 areas of the Venn Diagram clearly with a number and a label. Be sure that you have used the “both” number to make adjustments in the “side” numbers. All four numbers in your tree diagram should add to 200. If not, you will need to re-analyze your diagram.)

b) How many play both sports.

c) How many play basketball but not football?

d) How many play football but not basketball?

e) How many do not play football or basketball?

7) An architectural firm is putting out bids to design two large governmental buildings. Suppose they believe they have 35% chance of getting the contract for the first building, an 80% chance of getting the contract for the second building and a 10% chance of getting neither job.

a) Draw a Venn Diagram for this situation and use your diagram to find the chance that they get both contracts.

b) Use a formula for this situation to find the chance that they get both contracts.

8) A student tells their teacher that they want to build a cabinet in woodshop. Students sometimes build this project with oak only, sometimes with cherry only, sometimes with both and sometimes with neither. There is a 40% chance the project will be built using oak, a 50% chance the project will be built using cherry, and a 30% chance that the project will be built using both types of wood. What is the chance that the student will not use either oak or cherry?

9) Consider a set of 15 pool balls. Balls numbered 1 through 8 are solid and balls 9 through 15 are striped. Suppose the balls are placed into a bag and one ball is randomly selected. Find the probability that:

a) you selected either a solid ball or a ball numbered greater than 12?

b) you selected an even numbered ball or a solid ball?

c) you selected a solid ball or a striped ball?

d) you selected a ball that was striped and even?