module 7 part 2

A boy does not have any screen time each day and has a BMI of 18

The answer is A and B. The boy has a BMI between 15.4 and 25.2 and he has screen time between 0 and 75 minutes.

Click on the titles below to see the categories with the Venn diagram. A or B union

" A or B " is what in English we would normally define as " A and B ," that is everything that is in A and everything that is in B . You can remember the probability meaning of "or," the union, by realizing that the "or" means an outcome is in the category as long as it is in either A or B . The key words for this category are "or" and "either." A or B is called the union* of A and B . The total orange area indicates the union.

"Not"

The word "not" before a category is everything outside that category but within the universe. In the table below, "not A" is illustrated in orange. Remember A = "Bob wears a black suit." "Not A " is Bob wears a blue or brown suit. The color of the shoes is irrelevant.

Click on the titles below to see the categories with the Venn diagram. Not

The word "not" before a category is everything outside that category. In the Venn diagram above, the white space around A and the parts of B that are not in A represent "not A ," that is the outcomes that are not in A , but are still in the universe. "Not A " in our real-world example is "Americans employed last year as something other than factory workers." Teacher, postal worker, librarian, business woman, etc. all belong to "not A ." Note that it includes people in category B who are not in A , such as librarians who were injured by falling bookshelves. However, it is best not to explicitly consider B when trying to decide if something is not A . It's enough to ask "is this person someone who was employed last year in a type of factory work?". If not, she belongs to "not A ." In the diagram below, the orange shaded area represents all the elements in "not A .

The following tree graph will be used for Questions 5 through 7. The study is designed to learn whether healthy eating can affect heart disease. In order to measure the possible effects of healthy eating, we first need to calculate what the probability of being in a certain category would be assuming there is no effect, which is the same as assuming the probability of each outcome is equally likely. Use the / sign to indicate division. Always reduce fractions to lowest terms What is the probability that someone has inherited risk factors for heart disease (assuming all outcomes are equally likely)?

1/2 is Correct × The answer is 12. There are 6 possibilities for having risk factors of heart disease out of 12 total possibilities.

Similarly, if the experiment is flipping a coin, the events are "heads" or "tails." The probability of flipping a heads is 1/2 . The probability of flipping a tails is 1/2 . The sum of these two probabilities is

1/2+1/2= 2/2= 1

The following tree graph will be used for Questions 1 through 4. Use the / sign to indicate division. Always reduce fractions to lowest terms What is the probability of a family that has two girls with a boy in between?

1/8 is Correct × The answer is 18. There is only one way of having two girls with a boy in between. There are 8 equally likely outcomes. 18

For questions 5 - 7, use the tree method. On a piece of paper, construct a tree diagram to help you determine the sample space of the following situations, then type in the size of the sample space. 5. What is the size of the sample space for spinning a color on a spinner labeled red, yellow, and blue and then spinning a number on a spinner labeled 1 through 4?

12 is Correct × The answer is 12. From the tree diagram that was constructed, there are 12 outcomes.

There are 31 days in July. What is the probability of not selecting an even number day?

16/31 is Correct × The answer is 1631. There are 16 odd number days. Therefore, the probability is 1631.

There are 72 people under the age of 21 and 28 over the age of 21 at an event. What is the probability of not selecting someone over the age of 21 ?

18/25 is Correct × The answer is 1825. There are 72 + 28 = 100 people at the event. Therefore, the probability is 72100=1825.

Using the complement rule

1−P(A)=P(not A) 1−P(having no girls)=P(having at least one girl) 1−1/8=8/8−1/8=7/8 The probability that a family has at least one girl is 7/8 .

There are 10 blue, 10 green, and 10 yellow notecards. What is the probability of not selecting a green note card?

2/3 is Correct × The answer is 23. There are 10+10+10=30 notecards and 30−10=20 that are not green. Therefore, the probability is 2030=23.

The sum of all probabilities of individual outcomes in the sample space must equal 1 . True or False?

This is a true statement. The sum of all probabilities of individual outcomes equals 1 .

If the probability of the event given is 0 , then the complementary event will have probability equal to 1 . True or False?

This is a true statement. The sum of the probabilities of the event and its complement is 1 , so if the probability of the given event is zero, its complement has probability equal to 1 .

There are 32 houses with privacy fences, 45 houses with chain-link fences, and 23 with a different type of fence or no fences. What is the probability of selecting a house with a different type of fence or no fence?

23/100 is Correct × The answer is 23100. There are 32+45+23=10

What is the probability of having 3 kids that are not all boys or girls?

3/4 is Correct × The answer is 3/4. There are 8 possibilities for 3 kids. Two possibilities that are all boys or girls, so 6 are not are the same sex. Therefore, the probability is 68=34.

Roger can use either MLK Blvd or Pine Nut Ave. to get to intersection A. At that intersection, he can use one of three roads, Lilac St., Hunter St. or Ferret St. to get to work. How many different ways does Roger have to get to work?

6

There are 150 people at the restaurant and 80 are females. What is the probability of not selecting a female?

7/15 is Correct × The answer is 715. There are 150−80=70 males. Therefore, the probability is 70150=715.

There are 10 black, 6 blue, 12 yellow, and 4 red marbles in a bag. What is the probability of selecting a marble that is not red?

7/8 is Correct × The answer is 78. There are 10+6+12+4=32 marbles. There are 32−4=28 marbles that are not red. Therefore, the probability is 2832=78.

For questions 1-4 use the list method to determine your answer. On a piece of paper, construct a list to help you figure out the sample space of the following situations, then type in the size of the sample space. What is the size of the sample space for choosing an ice cream flavor (chocolate, strawberry, vanilla) and sauce (chocolate, caramel, or none)?

9 is Correct × The answer is 9. From the list that was constructed, there are 9 outcomes. Vanilla, Chocolate Vanilla, Caramel Vanilla, None Chocolate, Chocolate Chocolate, Caramel Chocolate, None Strawberry, Chocolate Strawberry, Caramel Strawberry, None

A and B (intersection)

A and B signals the set of elements where both A and B are true. It is called the intersection* of A and B. For Bob's outfits, A and B would be the situations in which Bob is wearing both a black suit and black shoes. A and B is represented below with orange cells.

A or B (union)

A or B is the category which includes all the outcomes in A, all the outcomes in B and all the outcomes in their intersection. It is called the union* of A or B. For Bob's outfits, A or B would be the situations in which Bob is wearing a black suit, black shoes, or both. A or B is represented below with orange cells.

A Tree

A tree is a way of writing a list that expands horizontally rather than vertically. It ensures that you do not miss an outcome. To create a tree to help you determine the sample space, start by writing the first set of options. Attach the next set of options to each item in the first set.

What is the probability that the family will have exactly 2 girls?

Again, count the number of rows that contain 2 girls and divide this number by 8 . The rows are boy-girl-girl girl-boy-girl and girl-girl-boy 3/8

Complementary events are

those that do not have any common outcomes, and the union is the whole universe. One outcome is the event itself, and the other outcome is the event not happening. For example, flipping a coin and it landing on either heads or tails — those are the only two possible outcomes, so the two events are complementary.

Not B refers to outcomes that are not in the category of B but still in the universe.

In our real-world example, not B would be employed people who were NOT injured at work last year, including all factory workers who were not injured at work, as well as all doctors, nurses, garbage workers, and truck drivers who were not injured at work. In the diagram below, the orange shaded area represents all the elements in "not B "

The fact that the probabilities of an event and its complement sum to 1 is not just interesting; it can be useful with some real-world problems. In some cases it may be difficult to calculate the probability of an event, but easier to calculate the probability of its complement. In these cases 1 − P(complement) might provide an easy calculation to a difficult problem

In some cases it may be difficult to calculate the probability of an event, but easier to calculate the probability of its complement. In these cases 1 − P(complement) might provide an easy calculation to a difficult problem

Sample Spaces and Probabilities

In the previous example of having three children, we should assume that having a boy is as likely as having a girl (though in practice having a boy is slightly more likely). Using our basic rule of probability, we find the likelihood by calculating the number of ways for the desired event to occur divided by the number of total outcomes. We can use the tree to calculate different probabilities. For the following probabilities, we are assuming a family has or is planning to have exactly three children.

For questions 1-4 use the list method to determine your answer. On a piece of paper, construct a list to help you figure out the sample space of the following situations, then type in the size of the sample space. What is the size of the sample space for rolling 1 die and flipping 1 coin?

The answer is 12. From the list that was constructed, there are 12 outcomes. 1, Heads 1, Tails 2, Heads 2, Tails 3, Heads 3, Tails 4, Heads 4, Tails 5, Heads 5, Tails 6, Heads 6, Tails

Using a tree diagram can result in probability greater than 1 . True or False?

This is a false statement. The probability of an event can never be greater than 1 .

What is the probability that someone has no risk factors and develops heart disease before the age of 50?

The answer is 16. There are 2 rows that contain "no risk factors" and "develops heart disease before the age of 50." Therefore the probability is 16

What is the probability of selecting a month that does NOT have 31 days?

The answer is 512. There are 12 months in a year and 7 months with 31 days (January, March, May, July, August, October, and December). There are 5 months that do not have 31 days. Therefore, the probability is 512.

All events in the sample space have the same relative frequency. True or False?

This is a false statement. The relative frequency of an event is how often it happens in practice. For example, we do not know if people who do not have inherited risk factors for heart disease decrease their risk of developing heart-related diseases by eating healthy until we gather data. It is reasonable to assume, however, that the relative frequency of different eating habits (healthy versus not-healthy) might not be equal.

The sample space and sample size represent the same thing. True or False?

This is a false statement. The sample space is a set of outcomes while the sample size is the number of different outcomes.

Complementary events occur whenever there are two events. True or False?

This is a false statement. The two events must be defined events.

If two events have no common outcomes, they are complementary. True of False?

This is a false statement. Two events with no common outcomes are complementary only if their union is the whole universe.

Any experiment or trial will have only one sample space. True or False?

This is a true statement. A sample space is the set of outcomes for one particular experiment, so any experiment has one unique sample space.

In a tree diagram showing the sample space of an experiment, each new option increases the sample size. True or False?

This is a true statement. Each new option will be added to each of the previous options, increasing the size of the sample space dramatically.

The sample space can have equally likely outcomes. True or False?

This is a true statement. Each outcome in a sample space can have the same chance of occurring.

Finding the probability of events, given all outcomes are equally likely, has limited applications. True or False?

This is a true statement. Most probabilities are calculated when there are not equally likely outcomes.

A list, table, or tree diagram will show the exact same sample space. True or False?

This is a true statement. No matter the method used, each technique will result in the same sample space.

The complement of an event occurring is the event not occurring. True or False?

This is a true statement. The opposite of an event happening is the event not occuring.

The probability using sample spaces with equally likely outcomes is the number of outcomes in the event divided by the sample size. True or false?

This is a true statement. The probability is the number of outcomes in the event divided by the sample size.

What is the probability that a family will have a boy and then a girl and then another boy?

This is one possibility out of the total of 8 different options. probability 1/8

Sample Space

This set of possible outcomes in an experiment is called the sample space*. The sample space of flipping a coin is heads and tails. The sample space of rolling a regular six-sided die is 1 , 2 , 3 , 4 , 5 , and 6

Category Concepts

Throughout this module, we have been primarily examining events comprised of one outcome. Now we will broaden our scope to include more complicated events. Consider a man named Bob who has five suits and three pairs of shoes. He has one black suit, one brown suit, and three blue suits. He has two pairs of black shoes and one pair of brown shoes. We can put the two categories of clothes together. The table below lays out all the different possibilities of shoes and suits. black suit black suit, black shoes 1 black suit, black shoes 2 black suit, brown shoes brown suit brown suit, black shoes 1 brown suit, black shoes 2 brown suit, brown shoes blue suit 1 blue suit 1 , black shoes 1 blue suit 1 , black shoes 2 blue suit 1 , brown shoes blue suit 2 blue suit 2 , black shoes 1 blue suit 2 , black shoes 2 blue suit 2 , brown shoes blue suit 3 blue suit 3 , black shoes 1 blue suit 3 , black shoes 2 blue suit 3 , brown shoes Let A = Bob wears a black suit. Let B = Bob wears black shoes. We can now use this table to define some important categories in probability.

What is the probability that a family will have at least one boy?

To do this problem, count up the number of tree branches that contain at least one boy. That is every row except girl-girl-girl, so a total of 7 . 7/8

If there are outcomes in the universe that are not in either event, then

the two events are not complementary. For example, picking either a large shirt or a medium shirt are not complementary events. There are shirts in many sizes! You could pick a large, medium, small, extra large, etc. So picking a large shirt or a medium shirt are not complementary.

More generally, because "at least one X " and "no X " are complements, it is always true that

P(at least one X) = 1 − P(no X)

The events must be properly defined in order to have complementary events. For example,

, "rolling a die and getting an even number" would not be a complete instance of complementary events (as this is only one event). However, "rolling a die and getting an even number" and "rolling the die and getting an odd number" are complementary events; the events are clearly defined.

Sample Space of Having 2 Children Boy Boy Boy Girl Girl Boy Girl Girl

Boy Boy Boy Girl Girl Boy Girl Girl You can see from the table above that the sample size is 4 and there are two ways to have a boy and a girl. The probability of having 1 boy and 1 girl when having 2 children is 24=12 . When making a list, be sure to create as much of a pattern as you can to ensure you have exhausted all possibilities. For example, you might organize the list above by starting with the maximum number of boys you can get.

Now, let us try a more complicated example. Try to list all the possible ways to have three children in the box below. Make each item in the list its own line.

Boy Boy Boy Boy Boy Girl Boy Girl Boy Boy Girl Girl Girl Boy Boy Girl Boy Girl Girl Girl Boy Girl Girl Girl Notice that the list above creates a pattern of keeping the first outcome (boy) as long as possible. Without a lot of organization and concentration, it is easy to miss one outcome in the sample space. Other methods for discovering a sample space are more reliable.

Example Returning to Bob's suits:

If Bob likes the suits equally, what is the probability he will choose to wear blue? P(Bob wearing a blue suit)=(3 blue suits)(5 suits total) =35 What is the probability of Bob not wearing a blue suit? The probability of Bob not wearing a blue suit is equal to 1−P(Bob wearing a blue suit) . Since P(Bob wearing a blue suit) is 3/5 , the probability of Bob not wearing a blue suit can also be calculated as, 1−3/5=5/5−3/5=2/5 P(Bob not wearing a blue suit)=2/5

Click on the titles below to see the categories with the Venn diagram. A and B intersection

It is easy to get confused with the expression " A and B ." The word "and" in English usually indicates a more expansive possibility. You might say "don't bring just Jack to the party; bring Jack AND Jill!" In probability, on the other hand, the word "and" indicates greater restriction. We are considering only those elements in both A and B . This is why the keywords for this category are "and" and "both." The intersection of A and B will be the same size or smaller—not larger—than the sets of A or B individually. In our real-world example, we are considering only factory workers who suffered workplace related injuries. We are excluding factory workers who did not have any injuries and librarians who did. A and B includes fewer elements than either A or B alone. A and B is called the intersection* of the two categories. It is represented in pale purple below.

Something Must Happen It is important to remember that the probabilities of all the outcomes in the sample space always sum to

It is important to remember that the probabilities of all the outcomes in the sample space always sum to 1. That is, there is a 100 percent chance of one of the outcomes in the sample space happening, since the sample space, by definition, contains all possible outcomes.

Example: Probability of Complementary Events

Let A= the day is a weekday Let B= the day is a weekend What is P(A) ? What is P(B) ? The formula for probability is Number of desired events/Total number of events The number of weekdays is 5 The total number of days in the week is 7 The number of weekend days is 2 The total number of days in the week is 7 P(A)=5/7 P(B)=2/7 5/7+2/7=1 Notice from the example above that since the sum of the probability of an event and its complement is 1 , the probability of 1 minus the probability of that event is the probability of the complement.

A 20 year study creates different categories of 40 year old patients: those with a greater inherited risk of heart disease and those without; those who have healthy eating habits at the start of the study and those who do not; those who get heart disease before age 50 , after the age of 50 but before age 60 , and those who do not yet have heart disease at the end of the study.

List: inherited risk-healthy eating-heart disease before age 50 inherited risk-healthy eating-heart disease between 50 — 60 inherited risk-healthy eating-no heart disease by 60 inherited risk-unhealthy eating-heart disease before 50 inherited risk-unhealthy eating-heart disease between 50 — 60 inherited risk-unhealthy eating-no heart disease by 60 no inherited risk-healthy eating-heart disease before 50 no inherited risk-healthy eating-heart disease between 50 — 60 no inherited risk-healthy eating-no heart disease by 60 no inherited risk-unhealthy eating-heart disease before 50 no inherited risk-unhealthy eating-heart disease between 50 — 60 no inherited risk-unhealthy eating-no heart disease by 60 There are 12 different categories in the study

Venn Diagrams

Now let us consider the same categories using a more real-world example and a different visual aid: the Venn diagram that defines different categories spatially. For example, in the real world, we might ask what is the probability that a factory worker suffers a work-related injury in the past year? This question combines being a factory worker—a machinist, a textile worker, a structural iron and steel worker, or a mechanic (among many others)—and being a person who has suffered a work-related injury—a mailman slipping as he delivers mail, an assembler suffering repetitive stress injury, a construction worker falling from scaffolding (and many other injuries).

Complement Rule Part Two

One example of how the complement rule can be helpful is studying problems with the phrase "at least one." For example, remember our previous example of the family planning to have three children? There were eight possible outcomes in terms of gender. What is the probability that family will have "at least one" girl? The complement of "at least one X " is "no X ." The complement of having "at least one" girl is having "no girls." It turns out that calculating the probability of having "no girls" is very easy: it's the same as the probability of having three boys.

A List

One very straightforward way to determine a sample space is to write out all the options in a list. The downside of this method is it can be difficult to determine if you have missed an outcome. To show the method, suppose someone who is planning to have two children wants to know the probability of having one boy and one girl. To calculate the probability, you first must determine the sample space of possible outcomes. This is a relatively easy sample space to find.

Another way to write this is:

P(A)+P(not A)=1 1−P(A)=P(not A) 1−P(not A)=P(A)

Sum of Probabilities

Remember that the probabilities of each individual outcome in the sample space sum to 1. For example, suppose the experiment is "being born on a day of the week." The sample space would be Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday. The probability of being born on each of those days is 17 . Since there are 7 days, the sum of all the probabilities is 1/7+1/7+1/7+1/7+1/7+1/7+1/7 =7/7 =1

A tree diagram is the only method to determine the probability of an event. True or False?

This is a false statement. A list, table, tree diagram, and other methods can help determine the probability of an event.

The following tree graph will be used for Questions 1 through 4. Use the / sign to indicate division. Always reduce fractions to lowest terms . What is the probability that a family has at least two children of the same gender?

The answer is 1. There are eight ways of having a family with at least two children of the same gender (there are only 2 genders and there are 3 spots). There are 8 equally likely outcomes. 88. This fraction should be reduced to 1.

For questions 8 - 10, use the table method. On a piece of paper, construct a table to help you determine the sample space of the following situations, then type in the size of the sample space. You want an ice cream cone with one color of M&Ms as a topping. You pick one M&M color and one ice cream flavor at random. There are six M&M colors (brown, yellow, green, red, orange, and blue) and two ice cream flavors (chocolate and vanilla). What is the size of the sample space for the combinations of ice cream flavor and topping?

The answer is 12. From the table, we see there are 12 possible outcomes.

A six-sided die is rolled. What is the probability of not rolling a prime number?

The answer is 12. There are 3 prime numbers (2, 3, and 5) that can be rolled so 3 numbers are not prime. Therefore, the probability is 36=12.

What is the probability that someone in the study is a healthy eater?

The answer is 12. There are 6 rows that include healthy eating and 12 rows total. The answer is 12

The following tree graph will be used for Questions 1 through 4. Use the / sign to indicate division. Always reduce fractions to lowest terms What is the probability of a family that has two girls with a boy in between?

The answer is 14. There are 2 ways that a family does not have 2 children of the same gender in a row: boy-girl-boy and girl-boy-girl. There are 8 equally likely outcomes. 28. This fraction should be further reduced to 14.

For questions 5 - 7, use the tree method. On a piece of paper, construct a tree diagram to help you determine the sample space of the following situations, then type in the size of the sample space. . What is the size of the sample space for selecting a tree (maple, pine, oak), color of mulch (red, brown, black) and compost (yes or no)?

The answer is 18. From the tree diagram that was constructed, there are 18 outcomes.

For questions 8 - 10, use the table method. On a piece of paper, construct a table to help you determine the sample space of the following situations, then type in the size of the sample space. . What is the size of the sample space given that a number 1 through 10 is drawn from a bag and a coin is flipped?

The answer is 20. From the table, we see there are 20 outcomes.

For questions 8 - 10, use the table method. On a piece of paper, construct a table to help you determine the sample space of the following situations, then type in the size of the sample space. What is the size of the sample space for rolling a die (1 through 6) and spinning a spinner (1 through 4)?

The answer is 24. From the table, we see there are 24 outcomes.

The following tree graph will be used for Questions 1 through 4. Use the / sign to indicate division. Always reduce fractions to lowest terms What is the probability that a family has exactly one boy?

The answer is 38. There are three ways that a family can have exactly one boy. There are 8 equally likely outcomes. 38.

For questions 1-4 use the list method to determine your answer. On a piece of paper, construct a list to help you figure out the sample space of the following situations, then type in the size of the sample space. What is the size of the sample space for the order in which marbles are drawn out of a bag that contains 1 red, 1 yellow, and 1 green marble?

The answer is 6. From the list that was constructed, there are 6 outcomes. Red, Yellow, Green Red, Green, Yellow Yellow, Red, Green Yellow, Green, Red Green, Red, Yellow Green, Yellow, Red

. A person has 8 blue shirts, 10 white shirts, and 6 yellow shirts. What is the probability of not wearing a white shirt?

The answer is 712. There are 8+10+6=24 shirts. There are 24−10=14 shirts that are not white. Therefore, the probability is 1424=712.

For questions 1-4 use the list method to determine your answer. On a piece of paper, construct a list to help you figure out the sample space of the following situations, then type in the size of the sample space. What is the size of the sample space for flipping 3 coins?

The answer is 8. From the list constructed below, there are 8 outcomes. Heads, Heads, Heads Heads, Heads, Tails Heads, Tails, Heads Heads, Tails, Tails Tails, Tails, Tails Tails, Tails, Heads Tails, Heads, Tails Tails, Heads, Heads

For questions 5 - 7, use the tree method. On a piece of paper, construct a tree diagram to help you determine the sample space of the following situations, then type in the size of the sample space. What is the size of the sample space for selecting a fruit (banana, strawberry, orange) first, then a vegetable (carrots, celery, broccoli)?

The answer is 9. From the tree diagram that was constructed, there are 9 outcomes.

A boy with a BMI of 20 and 50 minutes of screen time.

The answer is A and B. The boy has a BMI of 20, which is between 15.4 and 25.2, so the event is in A. There is 50 minutes of screen time, which is between 0 and 75 minutes, so the event is in B. Therefore, A and B.

A boy spends 2 hours gaming and has a BMI of 24.1 .

The answer is A, but not B. The boy has a BMI of 24.1 which is between 15.4 and 25.2, so the event is in A. The screen time is 120 minutes, which is not between 0 and 75, so the event is not in B. Therefore, A but not B.

A boy texts his friends for three hours a day and has a BMI of 21.56 .

The answer is A, but not B. The boy's BMI falls inside the range of A, but his screen time falls outside the range of event B.

A boy spends 12 hour watching TV and has a BMI of 13 .

The answer is B, but not A. The boy has a BMI of 131 which is not between 15.4 and 25.2, so the event is not in A. The screen time is 30 minutes, which is between 0 and 75, so the event is in B. Therefore, B not A.

A boy has 18 minutes of screen time a day and has a BMI of 15 .

The answer is B, but not A. The boy has a BMI that falls outside the range of event A, but his screen time falls inside the range of event B.

. A boy has a BMI of 30 and has an hour of screen time each day.

The answer is B, but not A. The boy's BMI falls outside the BMI range, but he does have 60 minutes of screen time, putting him in event B.

Use the Venn diagram below to answer questions about what the areas represent. Suppose that: The universe is children's lives Event A is a boy having a BMI of 15.4 to 25.2 (the 5th to 95th percentile for 13 year old boys). Event B is spending between 0 and 75 minutes in front of a screen each day. The following events would fall into which areas? Your responses should be one of the following: (Enter the letter that corresponds with your answer choice) " A , but not B " " A and B " "Neither A nor B " " B , but not A " A girl has a BMI of 20 and spends 25 minutes in front of a TV each day

The answer is B, but not A. The girl is not a boy, so the event is not in A, but she does spend between 0 and 75 minutes in front of a screen, and so is in B.

A girl with a BMI of 22 and 100 minutes of screen time.

The answer is Neither A nor B. The girl is not a boy, so the event is not in A. The screen time is 100 minutes, which is not between 0 and 75 minutes so not in B. Therefore, neither A nor B.

A girl has a BMI of 20 and spends 135 minutes in front of a TV each day.

The answer is neither A nor B. The girl is not a boy and she spends an amount of time in front of the screen that is outside the range.

Complements

The opposite of an event happening (i.e. the event not happening) is called the complement* of the event. The sum of the probability of an event and the probability of its complement is always equal to 1 . For example, the complement of flipping heads is flipping tails. As we showed above the sum of flipping heads and flipping tails is 1 .

Once you have all the outcomes in the sample space listed, you read across the tree from left to right to get one unique outcome. The first "row" of this tree is boy-boy-boy, stretching upward and to the right. The second row is boy-boy-girl. You can also easily count the number of unique outcomes. Since each branch represents a unique outcome in the sample space, you count the last items.

The sample size of "different ways of having three children" is 8 Because the tree is systematic, you can be sure not to have missed an outcome.

A Table

The tree approach to finding the sample space also has limitations. It can become difficult to squeeze in every option if there are more than two. For more than two options, it is best to use a table. For a table, label the top and sides with the different options. Fill the cells in with the combination of options. The following is an example of a table illustrating the outcomes of a rolling two six sided die. The table again will help you list out every possible outcome in the sample space by organizing the outcomes of the two die rolls.

Universe

The universe is the entire category being considered. It encompasses both A and B and outcomes that are not in either category. In our example of Bob's shoes and suits, the universe would be Bob's outfits, the ensemble of suits and shoes. These outfits would not include shirts, ties, or socks! Every cell in the table would be included in the universe.

Click on the titles below to see the categories with the Venn diagram. Universe

The universe is the entire category being considered. It is not a given, but defined, just as the categories A and B are defined. In our example of factory workers in the past year ( A ) and people who have suffered from workplace injuries in the past year ( B ), the universe would be "people who were employed in the past year." The universe is defined based on the question that researchers are interested in studying. If we were calculating probabilities such as the probability of factory workers being injured at work compared to the probability of other workers being injured at work, it would make sense for the universe to be "all employed people within the past year." In the diagram below, the orange shaded area represents all the elements in the universe. Of course with "all people employed within the past year," as the universe, we could define other groups, such as "employed people who suffered injuries at home" or "employed people who made less than $20,000 a year." These might be important categories for other types of research questions.

What is the probability that a family will have 3 children of the same gender?

There are 2 ways to have children of the same gender: 3 boys or 3 girls probability 2/8 = 1/4

sample size*

While the sample space is all the different outcomes, the sample size* is the number of different outcomes. We need the sample space to determine the sample size, and we need the sample size to calculate probability. The sample size of flipping a coin is 2 ; the sample size of rolling a six-sided die is 6 .

The Venn diagram* can help convey

different categories. It shows a visual representation of all the possible results. The area labeled A represents A occurring—such as someone being a factory worker in the past year—while the area labeled B represents B happening—such being a person who suffered from a workplace injury in the past year. The darker area represents both A and B occurring, that is the set of all factory workers who suffered work-related injuries last year.

Each time you add options, make sure you include all of them for each of the earlier options. with a tree

You will notice that as you add more options, you will need to include them more times. The final row of "boy/girl" is added four times.