Knewton Alta Lesson 7 Part 1 Assignment
A bowl of candy contains 7 chocolate candies and 6 lemon candies. Choosing one piece of candy at random, find the probability of choosing a chocolate candy. Write your answer as a decimal, rounded to the hundredths place.
0.54 We need to find the probability of choosing a chocolate candy. Remember that probability is the number of favorable outcomes over the number of total possible outcomes. There are 7 favorable outcomes, because there are 7 chocolate candies in the bowl. There are 13 total possible outcomes (7 chocolate + 6 lemon equals a total of 13 candies). That means the probability of choosing a chocolate candy is 7 out of 13, or 7/13. Convert this fraction to a decimal by dividing the numerator by the denominator. The probability of choosing a chocolate candy is 0.54.
There are 26 cards in a hat, each of them containing a different letter of the alphabet. If one card is chosen at random, what is the probability that it is not between the letters L and P, inclusive? Write your answer in fraction form. Reduce the fraction if necessary.
21/26 Use the complement rule to find this probability. The complement of this event is the event that the card chosen has a letter from L to P. There are 5 of these letters so the probability of drawing a card with one of these letters is 5/26. The probability of the complement is 1−5/26= 26/26-5/26=21/26.
A bag contains 35 marbles, 11 of which are red. A marble is randomly selected from the bag, and it is blue. This blue marble is NOT placed back in the bag. A second marble is randomly drawn from the bag. Find the probability that this second marble is NOT red. Provide the final answer as a fraction.
23/34 Initially, 11 of the 35 marbles are red. That means that 35−11=24 of the marbles are NOT red. Once the blue marble is removed, the bag contains only 34 total marbles, but all 11 red marbles are still in the bag. Thus there are now 34−11=23 marbles that are NOT red. Thus, the probability that the second marble is NOT red is 23/34.
A single card is randomly drawn from a standard 52-card deck. Find the probability that the card is a face card AND is red.(Note: aces are not generally considered face cards, so there are 12 face cards. Also, a standard deck of cards is half red and half black.) Provide the final answer as a fraction.
3/26 There are 4 suits in a standard deck of cards, each suit containing 3 face cards (the jack, queen, and king). Thus, there are a total of 3⋅4=12 face cards. Of these 12 cards, half are red, which means there are 12÷2=6 red face cards. Since the entire deck contains 52 cards, the answer is 6/52, or 3/26. Another approach to this problem is to list each of the 6 red face cards: the jack, queen, and king of hearts, and the jack, queen, and king of diamonds. Again, since the deck contains a total of 52 cards, probability of drawing a red face card is 6/52, or 3/26.
A deck of cards contains RED cards numbered 1,2,3,4,5, BLUE cards numbered 1,2,3,4, and GREEN cards numbered 1,2,3,4,5,6. If a single card is picked at random, what is the probability that the card is GREEN? Select the correct answer below: 10/15 6/15 11/15 8/15 4/15 13/15
6/15 Because there are 6 green cards, and 15 cards total in the deck, the probability is 6/15.
A moving company has boxes numbered 1 through 40. What is the probability that the first box chosen by a mover is not a multiple of 9? Give your answer as a fraction. Reduce the fraction if necessary.
9/10 Note that there are 4 multiples of 9 in the range 1 to 40, namely 9, 18, 27, 36. So the probability of the mover picking a box of one of these multiples is 440. Therefore, by the complement rule, the probability of not picking a box numbered with a multiple of 9 is 1−4/40= 40/40-4/10=36/40=9/10.
A lawyer has numbered the cases that he is working on. He has criminal cases numbered 1,2,3,4 and civil numbered 1,2,3,4,5. Let R be the event of selecting a criminal case, C the event of selecting a civil case, E the event of selecting an even numbered case, and O the event of selecting an odd case.Selecting the criminal case number 4 is one of the outcomes in which of the following events? Select all correct answers. Select all that apply: C′ R AND O C OR O C AND O C AND E R AND E
C′ R AND E Because the case is criminal and the number is even, the case is an outcome of R and E. Therefore, it is also an outcome of R AND E and C′.
A track team contains sprinters numbered 1,2,3,4,5 and distance runners numbered 1,2. Let S be the event of selecting a sprinter, D the event of selecting a distance runner, E the event of selecting an even numbered runner, and O the event of selecting an odd runner. Selecting the sprinter numbered 1 is one outcome of which of the following events? Select all correct answers. Select all that apply: D AND O D OR E D OR O O′ E′ S AND E
D OR O E′ Because the runner is a sprinter and the number is odd, the runner is an example of S and O. Therefore, it is also an example of E′ (not even) and D OR O.
A standard six-sided die shows a number, 1, 2, 3, 4, 5, or 6, on each of its sides. You roll the die once. Let E be the event of rolling the die and it showing an even number on top and L be the event of rolling a number less than 4. Rolling a 3 is an outcome of which of the following events? Select all correct answers. Select all that apply: E OR L E′ AND L E AND L E′ OR L′ L′ E AND L′
E OR L E′ AND L E′ OR L′ To roll a 3: It is an outcome of E′, that is NOT even. It is an outcome of L, less than 4. So, correct the only correct "AND" answer is: E′ AND L. There are many more correct "OR" answers: E′∪L E′∪L′ E∪L
A biologist has a number of butterfly specimens. The butterflies are of various colors and various ages. The colors are green (abbreviated G), red (abbreviated R), or yellow (abbreviated Y). Each specimen is labeled with one the numbers {1,2,3,4,5,6} and the number represents how many months old it is. The dots in the Venn diagram below show the age and the color of the specimen. The biologist selects a specimen at random. Let A be the event of selecting a yellow specimen. Let B be the event of selecting a specimen that is older than 2 months old. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
Event A is the event of selecting a yellow butterfly, so A should contain all butterflies with a Y. Event B is the event of selecting a butterfly older than two months, so B contains all butterflies labeled with a 3 or higher for any color. Event A AND B should therefore contain all outcomes that are greater than 2 AND yellow. Therefore, A AND B does not contain any outcomes. Notice that the outcomes 1 and 2 on green and red butterflies do not fall into either of these events. They should therefore be outside of the Venn diagram.
A CEO decides to award her employees that have met their objectives this year. Those employees that have met their objectives have the chance to win vacation days. They can win either Mondays (abbreviated M) or Tuesdays (abbreviated T). They can also win up to two days. The Venn Diagrams below show the different combinations that an employee can win. Let A be the event of winning two of the same day. Let B be the event of winning a Monday Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
Event A is the event of winning two of the same day, so A should contain elements with repeated letters. Event B is the event of winning at least one Monday, so B should contain outcomes with the letter M . Event A AND B should therefore contain the outcome with two Mondays. Notice that the outcomes T does not fall into either of these events. It should therefore be outside of the Venn diagram.
Let C be the event that a randomly chosen cancer patient has received chemotherapy. Let E be the event that a randomly chosen cancer patient has received elective surgery. Identify the answer which expresses the following with correct notation: Of all the cancer patients that have received chemotherapy, the probability that a randomly chosen cancer patient has had elective surgery. Select the correct answer below: P(C|E) P(E|C) P(E) AND P(C) P(C AND E)
P(E|C) Remember that in general, P(A|B) is read as "The probability of A given B," or equivalently, as "Of all the times B occurs, the probability that A occurs also." So in this case, the phrase "Of all the cancer patients who have received chemotherapy" can be rephrased to mean "Given that a cancer patient has received chemotherapy," so the correct answer is P(E|C).
A car dealership finds that a certain model of new car has something wrong with its transmission 15% of the time. How likely is it that a particular model of that car has something wrong with its transmission? Select the correct answer below: Very likely, the probability is close to 1. Somewhat likely, the probability is closer to 1 than to 0. Unlikely, the probability is close to 0. Somewhat unlikely, the probability is closer to 0 than it is to 1. Equally likely, the probability is 0.5.
Somewhat unlikely, the probability is closer to 0 than it is to 1. Here, we would be most correct to say that the probability is somewhat unlikely . It is not closer to 1 than it is to 0, so it is not appropriate to call it "likely" or "somewhat likely." It is not very close to 0, so it is not quite appropriate to call it "unlikely". It is false to call it "equally likely" since the probability isn't 0.5. We settle on "somewhat unlikely," since it is closer to 0 than to 1 but isn't that close to zero.
A computer randomly generates numbers 1 through 100 for a lottery game. Every lottery ticket has 7 numbers on it. Identify the correct experiment, trial, and outcome below: Select all that apply: The experiment is the computer randomly generating a number. The experiment is the computer randomly generating a number less than 10. A trial is one number generated. The trial is identifying the number generated. An outcome is the number 2 being generated. The outcome is the number being randomly generated.
The experiment is the computer randomly generating a number. A trial is one number generated. An outcome is the number 2 being generated. An experiment is a planned operation carried out under controlled conditions. Here, generating a random number is the experiment. A trial is one instance of a an experiment taking place. A trial here is one number being generated. An outcome is any of the possible results of the experiment. Here, the outcome is any of the numbers between 1 and 100.
Using a standard 52-card deck, Michelle will draw 6 cards with replacement. If Event A = drawing all hearts and Event B = drawing no face cards, which of the following best describes events A and B? Select the correct answer below: independent dependent mutually exclusive complement
independent Events are independent if the knowledge of one event occurring does not affect the chance of the other event occurring. In this case, the chance of either event occurring is the same whether or not the other event occurs.
Seventy cards are numbered 1 through 70, one number per card. One card is randomly selected from the deck. What is the probability that the number drawn is a multiple of 3 AND a multiple of 5? Enter your answer as a simplified fraction.
2/35 From the first 70 natural numbers, there are 23 multiples are 3: 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66, and 69. There are 14 multiples of 5: 5,10,15,20,25,30,35,40,45,50,55,60,65, and 70. The intersection of these two sets contains the four numbers: 15,30,45, and 60. Thus, the probability that the number is a multiple of 3 AND a multiple of 5 is 470, which simplifies to 235. ALTERNATIVE SOLUTION: Any number that is a multiple of both 3 and 5 is a multiple of 3⋅5=15. Thus, one can simply count the multiples of 15 that are less than or equal to 70. Those numbers are 15,30,45, and 60, which brings us to the same conclusion of 4/70, or 2/35.
A deck of cards contains RED cards numbered 1,2,3,4,5,6, BLUE cards numbered 1,2,3,4,5, and GREEN cards numbered 1,2,3. If a single card is picked at random, what is the probability that the card is RED? Select the correct answer below: 11/14 3/14 1/14 4/14 6/14 10/14
6/14 Because there are 6 red cards, and 14 cards total in the deck, the probability is 6/14.
Which of the following gives the definition of event? Select the correct answer below: the set of all possible outcomes of an experiment a subset of the set of all outcomes of an experiment a planned activity carried out under controlled conditions one specific execution of an experiment
a subset of the set of all outcomes of an experiment An event is defined as a subset of the set of all outcomes of an experiment.
Patricia will draw 8 cards from a standard 52-card deck with replacement. Which of the following are not events in this experiment? Select all that apply: drawing 8 hearts drawing 8 diamonds drawing 1 card drawing 4 aces and 4 kings
drawing 1 card Events are any combinations of outcomes or particular results in an experiment. Drawing 8 cards is the experiment and drawing 1 card is a trial of the experiment, neither of which specify a result or outcome.
Martin will draw 3 cards from a standard 52-card deck without replacement 5 different times. For each 3-card draw, he will record the number of red cards and the number of black cards. What is a trial of this experiment? Select the correct answer below: drawing 5 cards drawing 1 card drawing 3 cards drawing 15 cards
drawing 3 cards A trial is one specific execution of an experiment. In this case, each trial is a 3-card draw.
Trial best fits which of the following descriptions? Select the correct answer below: a particular result of an experiment a subset of the set of all outcomes of an experiment one repetition or instance of an experiment the set of all possible outcomes of an experiment
one repetition or instance of an experiment A trial is defined as one repetition or instance of an experiment.
Arianna will roll a standard die 10 times in which she will record the value of each roll. What is a trial of this experiment? Select the correct answer below: one roll of the die rolling at least one 5 ten rolls of the die rolling a sum of 40
one roll of the die A trial is one specific execution of an experiment. In this case, each trial is one roll of the die.
A bag contains 8 red beads, 3 blue beads, and 9 green beads. If a single bead is picked at random, what is the probability that the bead is blue? Select the correct answer below: 8/20 12/20 9/20 17/20 11/20 3/20
3/20 There are 3 blue beads, and the total number of beads is 8+3+9=20. So the probability of getting a blue bead is 320.
A bag contains 11 RED beads, 10 BLUE beads, and 4 GREEN beads. If a single bead is picked at random, what is the probability that the bead is GREEN? Select the correct answer below: 14/25 11/25 4/25 10/25 15/25 21/25
4/25 There are 4 green beads, and the total number of beads is 11+10+4=25. So the probability of getting a green bead is 4/25.
A deck of cards contains RED cards numbered 1,2,3, BLUE cards numbered 1,2,3,4, and GREEN cards numbered 1,2. If a single card is picked at random, what is the probability that the card has an ODD number? Select the correct answer below: 7/9 6/9 2/9 5/9 8/9 4/9
5/9 By counting, we can see that there are 5 odd cards, and a total of 9 cards in the deck. So the probability is 5/9.
A deck of cards contains RED cards numbered 1,2,3,4,5, BLUE cards numbered 1,2,3,4,5,6, and GREEN cards numbered 1,2. If a single card is picked at random, what is the probability that the card is RED? Select the correct answer below: 8/13 2/13 5/13 3/13 1/13 6/13
5/13 Because there are 5 red cards, and 13 cards total in the deck, the probability is 5/13.
An HR director numbered employees 1 through 50 for an extra vacation day contest. What is the probability that the HR director will select an employee who is not a multiple of 13? Give your answer as a fraction. Reduce the fraction if necessary.
47/50 Note that there are 3 multiples of 13 in the range 1 to 50, namely 13, 26, 39. So the probability of the HR director selecting one of these multiples is 350. Therefore, by the complement rule, the probability of not selecting a multiple of 13 is 1−3/50= 50/50-3/50=47/50.
Phones collected from a conferences are labeled 1 through 40. What is the probability that the conference speaker will choose a number that is not a multiple of 6? Give your answer as a fraction. Reduce the fraction if necessary.
17/20 Note that there are 6 multiples of 6 in the range 1 to 40, namely 6, 12, 18, 24, 30, 36. So the probability of the speaker choosing one of these multiples is 6/40. Therefore, by the complement rule, the probability of not choosing a multiple of 6 is 1−6/40= 40/40-6/40- 34/40=17/20.
A university offers finance courses numbered 1,2,3,4,5 and accounting courses numbered 1,2,3,4,5,6. Let F be the event of selecting a finance course, A the event of selecting an accounting course, E the event of selecting an even numbered course, and O the event of selecting an odd course. Selecting the accounting course number 3 is one of the outcomes in which of the following events? Select all correct answers. Select all that apply: A AND O F OR E F AND O F OR O E′ A′
A AND O F OR O E′ Because the course is accounting and the number is odd, the course is an outcome of A and O. Therefore, it is also an outcome of A AND O, E′, and F OR O.
A deck of cards contains RED cards numbered 1,2,3, BLUE cards numbered 1,2,3,4, and GREEN cards numbered 1,2. If a single card is picked at random, what is the probability that the card is BLUE OR has an ODD number? Provide the final answer as a fraction.
7/9 There are a total of 3+4+2=9 cards. Of these, there are 4 blue cards, 2 red odd cards, and 1 green odd card. Thus the total number of cards we are interested in is 7, so the answer is 7/9.
A deck of cards contains RED cards numbered 1,2 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card. Drawing the Blue 1 is one of the outcomes in which of the following events? Select all correct answers. R AND E R AND O B AND O R′ B AND E B′
B AND O R′ Because the card is blue and the number is odd, the card is an outcome of B and O. Therefore, it is also an outcome of B AND O and R′.
A deck of cards contains RED cards numbered 1,2,3 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card. Drawing the Red 1 is one of the outcomes in which of the following events? Select all correct answers. Select all that apply: R′ B′ O′ R OR E B OR O B OR E
B′ R OR E B OR O Because the card is red and the number is odd, the card is an outcome of R and O. Therefore, it is also an outcome of B′, R OR E, and B OR O.
A mathematics professor is organizing her classroom into groups for the final project. Each student will either be working on a graphing (G) project or writing a paper (P). Also, each student will be working on an economics (E), finance (F), sociology (S), or criminal justice (C) problem. The dots in the Venn diagram below show the different scenarios. Let A be the event of a student working on a graphing project. Let B be the event of a student writing a paper. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
Event A is the event of a student working on a graphing project, so A should contain all outcomes with a G. Event B is the event of a student writing a paper, so B should contain all outcomes with a P. Event A AND B should therefore contain all outcomes with both a P and a G; however, none exist. Notice also that nothing should be outside of the Venn Diagram because every project is either a graphing project or a paper.
Two fair dice are rolled, one blue, (abbreviated B) and one red, (abbreviated R). Each die has one of the numbers {1,2,3,4,5,6} on each of its faces. The dots in the Venn diagram below show the number and the color of the dice. Let A be the event of rolling an even number on either of the dice. Let B be the event of rolling a number greater than 4 on either of the dice. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
Event A is the event of rolling an even number on either of the dice, so A should contain elements {R2,B2,R4,B4,R6,B6}. Event B is the event of rolling a number greater than 4 on either of the dice, so B should contain outcomes {R5,B5,R6,B6}. Event A AND B should therefore contain all outcomes that are greater than 4 AND even. Therefore, A AND B should contain the outcomes {R6,B6}. Notice that the outcomes R1, B1, R3, and B3 do not fall into either of these events. They should therefore be outside of the Venn diagram.
Different types of advertising methods are being considered for a company's new product: a magazine ad (M), a television ad (T), a newspaper coupon (N), a radio ad (R), a coupon mailer (C), and a social media ad (S). The coupons are both good for ten dollars off the item. The dots in the Venn diagram below show the various methods. An advertising specialist considers a method at random to review. Let A be the event of selecting a method that gives a discount. Let B be the event of selecting a printed method. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots. Provide your answer below:
Event A is the event of selecting a method that gives a discount, so this includes both types of coupons. Event B is the event of selecting a printed method, so this includes the newspaper coupon, magazine ad, and coupon mailer. Event A AND B should therefore contain all printed methods that give a discount, so this includes the newspaper coupon and coupon mailer, which is all of event A. Notice that the social media ad, radio ad, and television ad do not fall into either of these events. They should therefore be outside of the Venn diagram.
Katy is deciding which charity to donate to. She is going to donate fifty dollars to each charity chosen, and she can donate to a women's shelter (abbreviated B), a charity for rescue animals (abbreviated R), and a children's foundation (C). The dots in the Venn diagram below show the combinations that she can donate to. Let X be the event of donating fifty dollars each to two charities for a total of one hundred dollars. Let Y be the event of donating to the charity for rescue animals. Move the dots on the Venn diagram to place the dots in the correct event, X, Y,or X AND Y. Note that you might not use all of the dots.
Event X is the event of donating one hundred dollars, which means 50 dollars to two charities, so X should contain outcomes BR, RC, and BC. Event Y is the event of donating to the charity for rescue animals, so Y should contain any outcome with an R. Event X AND Y should therefore contain all outcomes that have two charities listed and an R. Therefore, X AND Y should contain the outcome BR and RC. Notice that the outcomes B and C do not fall into either of these events. They should therefore be outside of the Venn diagram.
A deck of cards contains RED cards numbered 1,2,3,4,5,6 and BLUE cards numbered 1,2,3,4,5. Let: R be the event of drawing a red card, B be the event of drawing a blue card, E be the event of drawing an even numbered card, and O be the event of drawing an odd card. Drawing the Blue 3 is one of the outcomes in which of the following events? Select all that apply: B′ R OR E R AND O E′ O′ B AND O
E′ B AND O Because the card is blue and the number is odd, the card is an outcome of B and O. Since the card is not even, it is an element of E′ (remember that E′ is the complement of E).Therefore, it is also an outcome of B AND O and E′.
A deck of cards contains RED cards numbered 1,2,3,4,5 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card. Drawing the Red 5 is one of the outcomes in which of the following events? Select all correct answers. Select all that apply: E′ R OR E B AND O B AND E R AND O O′
E′ R OR E R AND O Because the card is red and the number is odd, the card is an outcome of R and O. Therefore, it is also an outcome of E′, R AND O, and R OR E.
Paul will roll two standard dice simultaneously. If Event A = both dice are odd and Event B = at least one die is even, which of the following best describes events A and B? Select two answers. Select all that apply: Mutually Exclusive Not Mutually Exclusive Independent Dependent
Mutually Exclusive Dependent Mutually exclusive events are events that cannot occur at the same time. In this case, both dice cannot be odd (event A) if at least one of the dice is even (event B). Independent events are those for the occurrence of one event has no effect on the probability of the other, and dependent events are any that are not independent. Mutually exclusive events are almost always also dependent (with the exception of events that are already impossible) since the occurrence of one event means the probability of the other event changes 0.
Let B be the event that a randomly chosen person has low blood pressure. Let E be the event that a randomly chosen person exercises regularly. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen person exercises regularly, given that the person has low blood pressure. Select the correct answer below: P(E|B) P(B|E) P(E) AND P(B) P(B AND E)
P(E|B) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the person has low blood pressure, so the correct answer is P(E|B).
Let S be the event that a randomly chosen store is having a sale. Let M be the event that a randomly chosen store has marked up their prices in the last six months. Identify the answer which expresses the following with correct notation: Given that the store is having a sale, the probability that a randomly chosen store has marked up their prices in the last six months. Select the correct answer below: P(M) AND P(S) P(M|S) P(S|M) P(S AND M)
P(M|S) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the store is having a sale, so the correct answer is P(M|S).
Let S be the event that a randomly chosen voter supports the president. Let W be the event that a randomly chosen voter is a woman. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen voter is a woman, given that the voter supports the president. Select the correct answer below: P(W|S) P(S AND W) P(W) AND P(S) P(S|W)
P(W|S) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the voter supports the president, so the correct answer is P(W|S).
A deck of cards contains RED cards numbered 1,2,3,4 and BLUE cards numbered 1,2,3, as shown below. Blue cards: 1, 2, and 3 Red cards: 1, 2, 3, 4. Let R be the event of drawing a red card, B be the event of drawing a blue card, E be the event of drawing an even numbered card, and O be the event of drawing an odd numbered card. Drawing the Red 3 is an outcome in which of the following events? Select all correct answers. Select all that apply: R AND O B OR E R′ E′ E OR R
R AND O E′ E OR R The Red 3 is both red and odd, so it is an outcome in both R and O. Therefore, it is an outcome in R AND O. The Red 3 is not even, so it is an outcome in E′. The Red 3 is an outcome in R, so it is in E OR R (even though it is not in E).
