MA 132 Use of the Probability Model on Frequencies of Colors Problems I have handout and it has questions and I will attach the slides and the definition y

MA 132 Use of the Probability Model on Frequencies of Colors Problems I have handout and it has questions and I will attach the slides and the definition you need to answer the questions and I need it on word only and put the answers under each question Section 5.1 Questions
1. Determine which of the following tables of frequencies of colors of M&Ms in a bag is a probability model. For tables that are not probability models, explain why not.
2. In the bag of M&Ms represented by Table 1, which of the following is true? (Circle ALL that
apply)
(a) There are no blue M&Ms in the bag.
(b) All the M&Ms in the bag are blue.
(c) It is CERTAIN that you will choose a blue M&M when choosing an M&M from this bag.
(d) It is IMPOSSIBLE to choose a blue M&M when choosing an M&M from this bag.
3. In the bag of M&Ms represented by Table 3, which of the following is true? (Circle ALL that
apply)
(a) There are no yellow M&Ms in the bag.
(b) All the M&Ms in the bag are yellow.
(c) It is CERTAIN that you will choose a yellow M&M when choosing an M&M from this
bag.
(d) It is IMPOSSIBLE to choose a yellow M&M when choosing an M&M from this bag.
4. In the bag of M&Ms represented by Table 1, are there any colors of M&Ms that would be
considered unusual to choose? Please explain your answer.
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5. The following table represents the results of a survey conducted by the Centers for Disease
Control to determine college students? health-risk behaviors. Students were asked “How often
do you wear a seatbelt when driving a car?”
Response
Frequency
Never Rarely
118
249
Sometimes Most of the Time
345
716
Always
3093
(a) Construct a probability model for seatbelt use by filling in the table below.
Response
Probability
Never
Rarely
Sometimes
Most of the Time
Always
(b) Is it unusual for a student to never wear a seatbelt when driving in a car? Why or why
not?
6. If a basketball player shoots 3 free throws, write the sample space of possible outcomes using
S for a made free throw and F for a missed free throw. The first 2 are done for you:
{ (S,S,S), (S,S,F),
(a) If all of the outcomes are equally likely, what is the probability that the basketball player
makes exactly 2 free throws?
(b) If all of the outcomes are equally likely, what is the probability that the basketball player
makes at least 2 free throws?
7. A survey of 500 randomly selected high school students determined that 288 played organized
sports. What is the probability that a randomly selected high school student plays organized
sports?
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Section 5.2 Questions
8. Let the sample space of an experiment be given by S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Let
the event E = {2, 3, 4, 5, 6, 7}. Let the event F = {5, 6, 7, 8, 9}. Let the event
G = {9, 10, 11, 12}.
(a) What is the probability of event E?
(b) What is the probability of event F ?
(c) List the outcomes that are in both E and G.
(d) Are E and G mutually exclusive?
(e) Determine P (E or G).
(f) List the outcomes that are in both F and G.
(g) Are F and G mutually exclusive?
(h) Determine P (F or G).
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9. Using the table below, answer the following questions:
(a) Determine the probability that a randomly selected fatal crash involved a male.
(b) Determine the probability that a randomly selected fatal crash occurred on a Sunday.
(c) Determine the probability that a randomly selected fatal crash did NOT occur on a
Sunday.
(d) Determine the probability that a randomly selected fatal crash occurred on a Sunday
OR on a Monday.
(e) Determine the probability that a randomly selected fatal crash occurred on a Sunday
AND involved a male driver.
(f) Determine the probability that a randomly selected fatal crash occurred on a Sunday
OR involved a male driver.
(g) Would it be unusual for a fatal crash to occur on a Wednesday AND involve a female
driver? Please explain your answer.
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Section 5.3 Questions
10. Are the following pairs of events dependent or independent?
(a) E = Speeding on the interstate
F = Being pulled over by a police officer
(b) E = You get a high score on the statistics exam
F = The Washington Nationals win a baseball game
(c) E = You have blue eyes
F = Your friend has blue eyes
(d) E = Your parents both have blue eyes
F = You and all your siblings have blue eyes
11. About 13% of the population is left-handed. If two people are randomly selected, what is the
probability that both are left-handed?
12. What is the probability of flipping 5 heads in a row when flipping a coin?
13. What is the probability that a family with 5 children has all boys?
14. Would it be unusual for a family with 5 children to have all boys?
15. What is the probability that a family with 5 children have at least 1 girl?
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Section 5.4 Questions
16. Using the table below, answer the following questions.
(a) Determine the probability that a randomly selected fatal crash involved a male.
(b) Determine the probability that a randomly selected fatal crash involved a male GIVEN
that it occurred on a Sunday.
(c) Determine the probability that a randomly selected fatal crash occurred on a Sunday
GIVEN that it involved a male.
(d) Determine the probability that a randomly selected fatal crash occurred on a Sunday.
(e) Determine the probability that a randomly selected fatal crash occurred on a Friday
GIVEN that it involved a female.
(f) Determine the probability that a randomly selected fatal crash involved a male GIVEN
that it occurred on a weekend.
(g) Determine the probability that a randomly selected fatal crash occurred on a weekend
GIVEN that it involved a male driver..
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MA 132: Chapter 5 – Probability
Terminology
• Probability is the science of chance behavior. Behavior is unpredictable in the short run, but
has regular and predictable pattern in the long run.
• Experiment: a controlled operation that yields a set of results
• Outcomes: possible results of an experiment
• The sample space, S, of a probability experiment is the collection of all possible outcomes
• Event: any collection of outcomes from a probability experiment
• Probability: relative frequency of an outcome over the long run
• An event E is
– impossible if P (E) = 0
– a certainty of P (E) = 1
– an unusual event if P (E) < 0.05 (this cutoff point can change, but we will assume it is this unless otherwise noted) • The Law of Large Numbers: As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. • A probability model lists the possible outcomes of an experiment and each outcome’s probability • Empirical approach: collect data and calculate probabilities afterwards (using relative frequency) • Classical approach: probabilities can be determined a priori • The complement of event E, denoted E C , is all outcomes in the sample space that are not outcomes in the event E • Events A and B are disjoint (or mutually exclusive) if they have no outcomes in common. • Events A and B are independent if the occurrence of one does not affect the occurrence of the other; otherwise, they are dependent • Conditional Probability: The probability that event A occurs, given that B has occurred Rules of Probability 1. For P (E), the probability of an event E, we have 0 ≤ P (E) ≤ 1 2. The sum of the probabilities of all outcomes must equal 1 3. Using the Empirical approach, P (E) = frequency of E number of trials in experiment 4. Using the Classical approach, P (E) = number of ways E can occur number of all possible outcomes 5. Complement Rule P (E does not occur) = P (E C ) = 1 − P (E) 6. General Addition Rules P (A or B) = P (A) + P (B) − P (A and B) 7. Multiplication Rule: If A and B are independent, P (A and B) = P (A)P (B) MA 132 Section 5.1 Probability Rules Example n n n n n n n n Suppose a bag of M&Ms contains the following: 5 red 11 blue 7 green 2 yellow 15 tan 13 brown 53 total Construct a Frequency Distribution of the colors of M&Ms Color Frequency Relative Frequency Red 5 .094 Blue 11 .208 Green 7 .132 Yellow 2 .038 Tan 15 .283 Brown 13 .245 Total 53 1.000 Probability n What is the probability that if we choose 1 M&M from the bag at random, we will choose a brown M&M? 0.245 brown ) P( = Definitions n An experiment is a repeatable process where the results are uncertain An outcome is one specific possible result The set of all possible outcomes is the sample space n Example n n q Experiment: roll a fair 6 sided die q One of the outcomes … roll a “4” q The sample space … roll a “1” or “2” or “3” or “4” or “5” or “6” 5- { I , 2,3 4 , , 5,6 } Definitions (Continued) n n An event is a collection of possible outcomes … we will use capital letters such as E for events Example (continued) q n One of the events: E = {roll an even number} = { 2,4 , 6} A probability model lists the possible outcomes of an experiment and each outcome’s probability Rules n n 0≤# $ ≤1 The sum of the probabilities of all the outcomes must equal 1 q If we examine all possible cases, one of them must happen Rules (Continued) n n n If an event is impossible, then its probability must be equal to 0 (i.e. it can never happen) If an event is a certainty, then its probability must be equal to 1 (i.e. it always happens) If an event is unusual, then it has a low probability of occurring q Typically, we say an event is unusual if 2 3 < 0.05 Probability of an Event n If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by frequency of E P (E ) » number of trials of the experiment ex) n sH¥ , PCH = foot This becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers) Example n n n We wish to determine what proportion of students at a certain school have type A blood We perform an experiment (a simple random sample!) with 200 students If 58 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 58/200 = .29 Example (continued) n If 9 of those students have type AB blood, then we would estimate that the proportion of students at this school with type AB blood is 9/300 = .03 ↳ 0.045 This would be an unusual event because the probability is < .05 - n 0.045 = L 0.05 Classical Method n n n The classical method applies to situations where all possible outcomes have the same probability This is also called equally likely outcomes Examples q q q Flipping a fair coin … two outcomes (heads and tails) … both equally likely Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … all equally likely Choosing one student out of 250 in a simple random sample … 250 outcomes … all equally likely Formula number of ways " can occur ! " = number of all possible outcomes n Example: If we roll 1 fair die, what is the probability that we roll a number greater than 4? - - q q q Sample space: 5=51,2 Event: E 'S 5,63 P(E)= 2 0.33 6= , 3,415,6 } Empirical Probability • 341.329 . . • I 95 093 035 008.329 . n What is the probability that a thrown pig lands on “side with dot”? n Would it be unusual to throw a “Leaning Jowler”? P Cleaning Yes Towler) it = . 008 unusual so . 05 Handout n Please answer the Section 5.1 questions on the handout MA 132 Section 5.2 The Addition Rule and Complements Definitions n n Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events Example n S={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} q E= a number less than or equal to 2 1,23=58,93 = q SO , F= a number greater than or equal to 8 Example (continued) n Find q P(E) = 3- = 0.30 10 2- +0.20 q P(F) q P(E or F) = 10 = = 0.50 Addition Rule for Disjoint Events n If E and F are disjoint (mutually exclusive) events, then ! " or % = ! " + !(%) Example Number of Rooms Probability n What is the probability of in Housing Unit a randomly selected One 0.010 housing unit has two or Two 0.032 three rooms? Three 0.093 3) 172 Four 0.176 =P (2) t 173) Five 0.219 0.032 t -0.093 Six 0.189 0.125 Seven 0.122 Eight 0.079 or = = Nine or more 0.080 Example Number of Rooms Probability n What is the probability of in Housing Unit a randomly selected One 0.010 housing unit has five or Two 0.032 six or seven rooms? Three 0.093 7) 6 Pls Four 0.176 Pls) t Pl 6) t PG ) Five 0.219 O 122 0.219 t O 189 t Six 0.189 0.53 Seven 0.122 Eight 0.079 or or = . . = = Nine or more 0.080 The General Addition Rule n For any two events E and F ! " or % = ! " + ! % − !(" and %) E -71 2,3 } PCE or Ft - PCE ) = - PCE ) ft f , F- t 53,4, s} = f - - Pceandpy f Example n Suppose that a pair of dice are thrown. q q q Let E = “the first die is a two” Let F = “the sum of the dice is less than or equal to 5” Find P(E or F) using the General Addition Rule PCE and F) PCE ) t PCF) PCE or F ) - = = It IT - 36 = If f- Complement of an Event n The complement of E, denoted ! " , is all outcomes in the sample space S that are not outcomes in the event E # !" = 1 − # ! Entire region The area outside the circle represents Ec Example n n According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog? ! do not own a dog = 1 − !(own a dog) = = I - . . 684 316 Example n The data to the right represent the travel time to work for 318,800 residents of Hartford County, CT. q What is the probability a randomly selected resident has a travel time of 90 or more minutes? 4895318800 = 0 . 015 Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes. more I P ( 90 ) Pf less than o) q = = = or - I - 0 . 0.015 985 Handout n Answer the Section 5.2 questions on the handout MA 132 Section 5.3 Independence and the Multiplication Rule Definitions n n Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F Two events are dependent if the occurrence of event E does affect the probability of event F Examples: Independent or Not? n Suppose you draw a card from a standard 52-card deck of cards and then roll a die. The events “draw a heart” and “roll an even number” are independent because the results of choosing a card do not impact the results of the die toss. n If we draw two cards from a deck (one at a time) q If we replace the first card before be draw the second, the events are independent q If we do not replace the first card, the events are dependent Multiplication Rule for Independent Events n If E and F are independent events, then ! " and & = ! " ( ! & Example n The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that two randomly selected 60 year old females will survive the year? q q The survival of the first female is independent of the survival of the second female. We also have that P(survive) = 0.99186. P (First survives and second survives ) = P (First survives )× P (Second survives ) = (0.99186)(0.99186) = 0.9838 Example n A manufacturer of exercise equipment knows that 10% of their products are defective. They also know that only 30% of their customers will actually use the equipment in the first year after it is purchased. If there is a one-year warranty on the equipment, what proportion of the customers will actually make a valid warranty claim? q We assume that the defectiveness of the equipment is independent of the use of the equipment P (defective and used ) = P (defective )× P (used ) = (0.10)(0.30) = 0.03 Example n The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that four randomly selected 60 year old females will survive the year? C. 99186) ( 99186 ) C. 99186 ) C. 99186 ) " = = 1.99186) . 96784 Example: At Least Probabilities n The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that at least one of 500 randomly selected 60 year old females will die during the course of the year? P ( none ) I ) least P ( at - - one = = I = I = I = - P( no of one the dies soo C 99186 ) . - - . . 0168 9832 Soo ) Handout n Answer the Section 5.3 questions on the handout Purchase answer to see full attachment