3.1a Introduction to probability

A motor company is manufacturing pickup trucks, sedans, minivans, SUVs, ATV, and motorcycles. The dots in the Venn diagram below show the type of each vehicle. A vehicle is selected at random. -Let A be the event of selecting a four-wheeled vehicle. -Let B be the event of selecting a pickup truck or a motorcycle. 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. Note: ATVs are four-wheeled vehicles.

-Event A is the event of selecting a vehicle with four wheels, so A should contain the pickup truck, sedan, ATV, minivan, and the SUV. -Event B is the event of selecting a pickup truck or a motorcycle. -Event A AND B should therefore contain a pickup truck because it has four wheels.

Flights from Miami for New York City are delayed 27% of the time. How likely is it that a particular traveler's flight from Miami to New York will be delayed?

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 nursing student is planning his schedule for next quarter. He needs to take three courses, and one must be Statistics. For his other courses, he can choose one of two science classes and one of three social sciences classes. What is the probability that a randomly chosen schedule has statistics, anatomy, and economics classes?

1/6 Let the desired event, a statistics, anatomy, and economics schedule, be SAE. There is 1 possible outcome that matches this event. There are 6 possible outcomes (1 math × 2 science × 3 social sciences =6). So the probability of the schedule having statistics, anatomy, and economics is: P(SAE)=1/6

A bowl of candy contains 8 chocolate candies and 8 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.

.50 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 8 favorable outcomes, because there are 8 chocolate candies in the bowl. There are 16 total possible outcomes (8 chocolate + 8 lemon equals a total of 16 candies). That means the probability of choosing a chocolate candy is 8 out of 16, or 8/16. Convert this fraction to a decimal by dividing the numerator by the denominator. The probability of choosing a chocolate candy is 0.50.

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.

An airline tracks each of its airplanes' stops for the day. A particular airplane can travel to one of the following cities for each of its stops: What is the probability that the stops include Boston and Chicago?

1/8

In a game show, contestants are given the opportunity to win a new car if they correctly choose the winning door three times in a row. In the first round, they must choose between 3 doors, one of which is labeled "win." In the second round and third rounds, they will choose from 2 doors, with one labeled "win." Their options are shown in the tree diagram below. They are blind folded, and are not given an opportunity to see the doors beforehand. What is the probability that a contestant will NOT win a new car?

11/12 To win the new car, a contestant must choose a "win" door three times in a row. We can see from the tree diagram that there is only one possibility of doing this, but there are 12 possible outcomes. So, there are 12−1=11 ways to NOT win. The probability of a contestant not winning is then 1112.

A deck of cards contains RED cards numbered 1,2 and BLUE cards numbered 1,2,3. 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.Which of the following events include the outcome of drawing a blue 1?

B OR E B AND O Because the card is blue and the number is odd, the card is an outcome of the event B and of the event O. Therefore, it is also an outcome of the events B AND O and of the event B OR E.

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.

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 restaurant is offering chicken specials numbered 1,2,3,4,5,6 and fish specials numbered 1,2,3,4,5. Let: C be the event of selecting a chicken special, F be the event of selecting a fish special, E be the event of selecting an even numbered special, and O be the event of selecting an odd special. Selecting the fish special number 3 is one of the outcomes in which of the following events?

E' F and O Because the special is fish and the number is odd, the selection is an example of F and O. Since the selection is not odd, it is an element of E′ (remember that E′ is the complement of E).Therefore, it is also an example of C 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.

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.

At a sports event, a fair coin is flipped to determine which team has possession of the ball to start. The coin has two sides, heads, (H), and tails, (T). Identify the correct experiment, trial, and outcome below: Select all that apply: The experiment is identifying whether a heads or tails is flipped. The experiment is flipping the coin. A trial is flipping a heads. A trial is one flip of the coin. An outcome is flipping a tails. An outcome is flipping a coin once.

The experiment is flipping the coin. A trial is one flip of the coin. An outcome is flipping a tails. An experiment is a planned operation carried out under controlled conditions. Here, flipping the coin is the experiment. A trial is one instance of a an experiment taking place. A trial here is one flip of the coin. An outcome is any of the possible results of the experiment. Here, the outcome is flipping a heads or flipping a tails.

The average emergency room cares for 300 patients each day. There are 20 nurses on duty at the emergency room each day, and they share the patient load equally. Suppose one patient is chosen at random. Identify the numbers of each of the following and enter the probability as a simplified fraction: Provide your answer below: There are __ patients in the sample space. There are __ patients for each nurse daily. P(N)=__, is the probability that a patient will be assigned to a specific nurse.

The sample space is the set of all possible outcomes of an experiment. Here, there are 300 possible patients. The event is a grouping of the outcomes in the sample space. Here, there are 15 possible patients that a nurse may have. The probability of a patient being assigned to a specific nurse, N, is the comparison of being assigned to that nurse's patients versus all possible patients: P(N)=15/300=1/20.

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 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.

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.

A company is planning its spring advertising campaign. Its marketing team is planning to mail out different types of advertisements, at random, to their town's residents each week. They have the following types of advertisements ready: How many different possible ways can a resident receive an advertisement?

18 We could draw a tree diagram here to see all the possible outcomes, or we can use the Fundamental Counting Principle and quickly calculate the number of outcomes. The Fundamental Counting Principle tells us that we multiple the number of choices each week by one another to get the total number of possibilities. Possibilities=Week 1×Week 2×Week 3 Possibilities=2×3×3=18

The real estate manager of a commercial building needs to fill three empty spaces. The options for each store are: All of the possible options for the three stores are shown in the tree diagram below. What is the probability that the group of stores will have a restaurant or a hardware store, or both? Enter your answer as a fraction.

2/3 There are 4 possible outcomes that include a restaurant, a hardware store, or both. There are 6 possible store combinations (Possibilities = 1 grocery × 3 for the second store × 2 for the third store = 6). So the probability of having a group of stores with a restaurant or a hardware store is: P(restaurant or a hardware store)=4/6=2/3

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.

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.

Each side of a fair, six-sided die has a certain number of dots on it, 1,2,3,4,5, or 6. The die is rolled by a board game player as part of their turn. Identify the correct experiment, trial, and outcome below: The experiment is identifying the number that is rolled on the die. The experiment is rolling the die. A trial is one rolling of the die. The trial is identifying the number on the die. The outcome is rolling the die. The outcome is the number that is rolled on the die.

The experiment is rolling the die. A trial is one rolling of the die. The outcome is the number that is rolled on the die. An experiment is a planned operation carried out under controlled conditions. Here, rolling the die is the experiment. A trial is one instance of a an experiment taking place. A trial here is one rolling of the die. An outcome is any of the possible results of the experiment. Here, the outcome is any of the numbers on the die.

There are 150 new employees at a tech company in Southern California. 30 new employees are randomly assigned to each orientation group: A,B,C,D, and E.Identify the numbers of each of the following:Provide your answer below:There are _______ new employees in the sample space.There are _______ new employees in each event.P(C)=________, is the probability that you choose new employee that has been assigned to orientation group C.

The sample space is the set of all possible outcomes of an experiment. Here, there are 150 possible employees. The event is a grouping of the outcomes in the sample space. Here, there are 30 employees in each event: A,B,C,D,E. The probability of a new employee being assigned to group C is the comparison of being assigned to group C versus all the possible groups: P(C)=30/150=1/5.

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.

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 or L E' and L E' or L'

A credit card company requires that its customers choose a 3 digit, numerical code to access their online account. Each digit should be chosen from 0,1,2,3,4,5,6,7,8 or 9. So there are 10 possibilities for each digit. How many different three digit codes are there?

We know, possibility of choosing each digit is 10. So, total number of three digit codes are : T= 10 x 10 x 10 T=1000 Therefore, their are 1000 different three digit codes. Hence, this is the required solution.