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Probability is a simple question of how likely it is for a particular outcome to occur. We always look to divide the number of favorable outcomes by the total number of outcomes. We cannot predict a specific outcome for a random event but the Law of Large Numbers allows us to make long term predictions of chance behavior. The rules of probability dictate that we pay attention to whether events are independent, outcomes are mutually exclusive, or whether replacement is used. We also must deal with conditional probabilities for situations in which a particular outcome is assumed to have occurred. To help organize situations, use Venn diagrams, tree diagrams, or contingency (2-way) tables. Having a clear understanding of situations and being organized when dealing with probability is critical to successful calculations.
Unit 2 Review Exercises
Unit 2 Review Exercises
1) Suppose that 57 of 110 students at a school are underclassmen (freshmen or sophomores) while the rest of the students are upperclassmen (juniors or seniors). Suppose three students are selected at random.
a) What is the probability that all three of the students are underclassmen?
b) What is the probability that all three students are upperclassmen?
c) What is the probability that there is at least one underclassman and at least one upperclassman in the group of three students?
2) Two 6-sided dice are rolled, one after the other. Find each probability.
a) P(total of 10 or more)
b) P(doubles)
c) P(total is even or less than 6)
d) P(an odd product)
e) P(first die is greater than second die)
f) P(a 6 or a three is showing on at least one die)
g) P(an odd total or a 2 is showing)
3) A pet store surveys his customers during the day and finds that 15 customers own dogs and 9 own cats. Included in these were 4 customers who owned both.
a) Draw a Venn diagram for this situation
b) How many total customers were surveyed?
c) Suppose one of these customers was selected at random. What is P(owned a dog)?
d) Suppose one of these customers was selected at random. What is P(own a dog|own a cat)?
e) Suppose one of these customers was selected at random. What is P(own a cat|own a dog)?
4) Suppose that 40% of all adults in a certain town are females and that 60% are males. In addition, 60% of the females hold full-time jobs while 80% of the males hold full-time jobs.
a) Draw and label a tree diagram to represent this situation.
b) What is the chance that a randomly selected person holds a full-time job?
5) For Halloween at my house, kids spin a spinner that has three equally marked spaces labeled 1, 2, and 3. The number they spin is the number of pieces of candy they get. In my bag, I start with 20 chocolate bars and 30 sugar bombs - all with identical packaging. Trick-or-Treaters pick randomly out of my bag after they spin. I only restock my candy bag after each child finishes picking all their candy.
[Figure1]
a) What is the chance that a trick-or-treater gets to pick three pieces of candy?
b) Suppose a trick-or-treater spins a three. What is the chance that they pick three sugar bombs?
c) Suppose a trick-or-treater spins a three. What is the chance that they pick three chocolate bars?
d) What is the chance that the trick-or-treater gets only one chocolate bar and nothing else?
e) What is the chance that the trick-or-treater gets exactly one chocolate bar and one sugar bomb?
6) Suppose the table below gives a breakdown of the ages and genders of the teachers at your school.
Find the probability that a randomly selected teacher is:
a) a male.
b) 39 years old or younger.
c) either a male or at least 50 years old.
d) from 30 to 39 years old given that they are a female.
e) a female given that they are at least 40 years old.
7) The 2-way table shown below shows the number of different types of automobiles produced by major manufacturers.
What is the probability that a randomly selected vehicle is:
a) a Ford?
b) a truck?
c) a van or a Toyota?
d) a car given that the vehicle is built by GM?
e) a Ford given that the vehicle is a truck?
8) Two cards are dealt from a standard 52 card deck without replacement. What is the probability that the two cards are both face cards?
9) A baseball player has a batting average of .250 which means that he averages one hit for every four times he comes to the plate. What is the probability that this player will end up with exactly 2 hits if he comes to the plate 3 times in a single game?
10) Two bags have an assortment of marbles in them. The first bag contains 11 black, 12 white, and 7 gold marbles. The second bag contains 9 black and 11 white marbles. One marble is randomly selected out of each bag.
a) Draw a tree diagram to represent this situation.
b) What is the probability that the two marbles are both black?
c) What is the probability that the two marbles are the same color?
[Figure2]
11) A special deck of cards contains only the eight red cards that are face cards or aces. Two cards are dealt off the top of the deck.
a) What is the probability that the two cards you end up with are both kings?
b) What is the probability that the two cards are of different value?
c)What is the probability that the two cards have the same value (two kings, two queens, etc...)?
d) What is the probability that the two cards are the same suit?
12) A bag contains ten red cubes numbered 1 through 10 and five green cubes numbered 1 through 5. Two cubes are pulled from the bag at random. What is the probability that the two cubes are:
a) both red?
b) both odd?
c) the same color?
d) the same value?
13) For a carnival game, a bag contains one $100 bills and nine $20 bills. You roll a single 6-sided die one time. If you roll a one or two you get to pull one bill out of the bag. If you roll a three, four, five, or six, you get to pull two bills out of the bag.
a) Draw a tree diagram for this situation.
b) Build a probability model for this situation.
b) What is the probability that you win exactly $120?
14) A burglar alarm system has three separate detection mechanisms it uses to detect an intruder. Suppose a skilled burglar has an 30% chance to get around the first part of the detection system, a 60% chance of getting around the second part of the system, and a 55% chance of getting around the third part of the system. Assume each part of the detection system is independent of the other parts of the system.
[Figure3]
a) What is the chance that the system does not detect the burglar?
b) Based upon your answer to part a), what must be the chance that the system does detect the burglar?
c) What is the chance that the burglar can get around exactly two of the three detection systems
15) On a basketball team, players can play at least one of three positions; guard, forward, or center. Suppose that 30 girls try out for the basketball team. During tryouts 13 girls indicate they can play guard only, 3 state they can play center only, 6 state they can play center or forward and the rest state they can play forward only. A player is selected at random.
a) Draw a Venn diagram for this situation.
b) What is the probability that the randomly selected player says they can play forward?
c) Given that the player indicates they can play forward, what is the probability they can also play center?
16) A girl is deciding what jewelry to wear as she gets ready for school. She has 5 bracelets, 6 rings, and 8 necklaces from which to choose.
a) In how many ways can she choose exactly one of each item to wear from the 19 available items?
b) If she decides to randomly select three pieces of jewelry, what is the probability that all three of the items she picks are exacty the same type of jewelry?
c) What is the probability that she picks exactly one bracelet, one ring, and one necklace if she randomly selects three pieces of jewelry?
[Figure4]
17) Your statistics teacher needs to select 3 students to help demonstrate an activity. Your class has 12 sophomores, 19 juniors, and 5 seniors in it. Your teacher makes a random selection of three students.
a) In how many ways can your teacher select three students from this class of 36 students?
b) What is the probability that all three students will be juniors?
c) What is the probability that exactly one student from each grade will be selected?
18) All football plays that an offense can run can be classified as a pass, run, or a kick. No play can ever be put into two categories.
a) If an offense completes two plays, will these two plays be independent of each other? Why or why not?
b) If the offense runs one play, are the possible outcomes (pass, run, or kick) mutually exclusive? Why or why not?
Image References
Image References
Coins http://coinauctionshelp.com
Pool Balls http://plutonius.aibrean.com
Scattegories Die http://ehow.com
October Calendar http://printablecalendars.resources2u.com
Roulette Wheel http://www.partypokersupplies.co.uk
Kids at Board http://teachers.greenville.k12.sc.us
Two Striped Pool Balls http://demo.physics.uiuc.edu
Ace and King of Spades http://www.123rf.com
Seat Belt http://sawmengzhi.blogspot.com
Graduation http://www.prlog.org
Aerosmith http://www.obit-mag.com
Cabinet http://www.renovation-headquarters.com
Cathedral in St. Paul, MN http://www.scenicreflections.com
Tree http://www.onenewsnow.com
$100 Bill http://onlinecurrencytradingfxcm.blogspot.com
Royal Flush http://www.artpoker.net
Straight http://www.findabet.co.uk
Turtle http://www.maine.gov
Trick or Treaters http://www.myremoteradio.com
Bag of Marbles http://www.worldwiseimports.com
Burglar http://www.emovingstorage.com
Jewelry Box http://www.123rf.com
Image Attributions
Image Attributions
[1]^ License: CC BY-NC 3.0
[2]^ License: CC BY-NC 3.0
[3]^ License: CC BY-NC 3.0
[4]^ License: CC BY-NC 3.0
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