probability of compound events assignment

Compound events

A compound event is an event that includes two or more simple events. Simple events are events that can have only one outcome, while compound events can have multiple different outcomes. Compound events can be made up of a number of independent events (events in which the outcome of one event has no effect on the probability of the other) or dependent events (events in which the outcome of one event affects the probability of another).

The concept of compound events is used for determining compound probabilities.

Compound probability

A compound probability is the probability of a compound event. Generally, it is the ratio of favorable outcomes to the total number of outcomes within the sample space of the compound event and can be calculated using one of two rules: the addition rule and the multiplication rule.

Addition rule

The addition rule can be used for compound events in which the simple events involved do not occur together. This is referred to as being "mutually exclusive." For example, a person cannot weigh 160 pounds and 162 pounds at the same time. They can only be one or the other, so the probability of someone being 160 pounds or 162 pounds is mutually exclusive.

To find the probability that any one of several mutually exclusive events occurs, use the addition rule, and add the probabilities of each event:

P(A or B) = P(A) + P(B)

Reference the table below, and find the probability that a randomly selected student will weigh 50 kilograms (~130 pounds) or more.

Given that X is the random variable representing a student's weight, the probability that a student weighs 50 kg or more can be written as:

P(X ≥ 50)

Since the weight of a student is a mutually exclusive event, the probability of a randomly selected student weighing 50 kg or more can be found using the addition rule by summing the probabilities of all the weights 50 kg or greater:

P(X ≥ 50) = P(50) + P(51) + P(52) + P(53) + P(54 or more)

P(X ≥ 50) = 0.17 + 0.14 + 0.09 + 0.07 + 0.09 = 0.56

There is a 56% chance of a randomly selected student weighing 50 kg or more.

For events that aren't mutually exclusive, the addition rule can still be used, but overlap between the various outcomes needs to be taken into account. For example, assume that students are male with a probability of 0.55, taller than 5'4" with a probability of 0.35, or both with a probability of 0.10. To find the probability that a student is either male or taller than 5'4", we can add the first two probabilities (0.55 + 0.35 = 0.90), but need to subtract the probability that they are both, otherwise these students would be counted twice. Given that P(A) is the probability that a student is male, and P(B) is the probability that the student is taller than 5'4":

P(A or B) = P(A) + P(B) - P(A and B)

Therefore, the probability that a student is either male or taller than 5'4" is:

0.55 + 0.35 - 0.10 = 0.80

Multiplication rule

Probability of independent events.

The probability of a compound event where the events are independent events can be found by multiplying the probabilities of each independent event that makes up the compound event. Given two events, A and B, with probabilities of P(A) and P(B), the probability of both events occurring is:

P(A and B) = P(A)·P(B)

What is the probability of a die that is rolled twice landing on 3 both times?

Each time the die is rolled, it constitutes an independent event, so the outcome of the roll of a die does not affect the outcome of subsequent rolls. Given that P(A) is the probability of the first roll landing on 3, and P(B) is the probability of the second roll landing on 3:

P(A and B) = P(A) · P(B) = 1/6 × 1/6 = 1/36 = 0.0278

There is approximately a 2.78% chance of a fair die landing on 3 both times in 2 rolls.

Probability of dependent events

The probability of a compound event where the events are dependent events can be found by first calculating the probability of the first event, then calculating the probability of the second event occurring given that the first has already occurred (the conditional probability of the second event given the first). Multiplying the probability of the first event by the conditional probability of the second event, given the first, results in the probability of both events occurring:

P(A and B) = P(A)·P(B|A)

As an example, assume that 20% of the students in a high school are seniors and that 40% of students in the high school have taken pre-calculus. If being a senior had no effect on whether or not a student had taken pre-calculus, then the probability of being a senior and having taken pre-calculus would be (0.20)·(0.40) = 0.08. However, if it is observed that being a senior increases the probability that a student has taken pre-calculus to 65%, the above probability would be incorrect. The 40% chance of a high school student having taken pre-calculus would need to be adjusted to take into account that a senior is more likely to have taken the course.

The probability that a student is both a senior and has taken pre-calculus is therefore the probability of a student being a senior multiplied by the probability of the student having taken pre-calculus given that they are a senior:

P(senior and pre-calculus) = P(senior)·P(pre-calculus|senior)

P(senior and pre-calculus) = (0.20)·(0.65) = 0.13

Therefore, there is actually a 13% chance that a student is both a senior and has taken pre-calculus, rather than an 8% chance, since the latter doesn't take into account the increased probability of having taken pre-calculus by the time the student is a senior.

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Lesson Explainer: Compound Events Mathematics

In this explainer, we will learn how to find and interpret the probability of compound events.

In probability terms, when we speak of a simple event, we are looking for the probability that a single event occurs, for example, getting “heads” with a single toss of a coin. There are two sides to a coin, only one of which is a heads. The probability of getting “heads” is, therefore,

Recall also that probabilities can be written as either fractions, decimals (between 0 and 1), or percentages (between 1 % and 1 0 0 % ). So, with a single toss of a coin, the probability of getting “heads” is 1 2 = 0 . 5 = 0 . 5 × 1 0 0 % = 5 0 % .

Compound probability is concerned with finding the probability of more than one event occurring. Let us look at an example.

Example 1: Compound Probability: Throwing a Coin Twice

If we throw a coin twice, what is the probability of getting “heads” both times?

We are considering a compound event since there are two events involved here: throwing a heads on the first throw and throwing a heads on the second throw of the coin. We can illustrate all the possible outcomes on a tree diagram to help us find the probability of getting “heads” both times. The first set of branches represents the first throw, with the two possible outcomes “heads” and “tails.”

The probabilities for “heads” and “tails” are each 0.5. Now if we attach branches to each of these outcomes, covering all possible outcomes for the second throw, we can work out the compound probability of getting “heads” both times.

Using the same unbiased coin, the probability of getting “heads” on our second throw has not changed from 1 2 or 0.5. And we obtain the compound probability of getting one “heads” after another by multiplying the probability of “heads” on the first throw with the probability of “heads” on the second throw. We can write this as 𝑃 ( ∩ ) = 0 . 5 × 0 . 5 = 0 . 2 5 . H e a d s H e a d s

Before looking at another example, let us remind ourselves of one or two probability concepts and rules that will be useful to us.

Some Probability Rules and Definitions

For any event 𝐴 , if 𝑃 ( 𝐴 ) is the probability of event 𝐴 occurring, then we have the following:

Rule 1 0 ≤ 𝑃 ( 𝐴 ) ≤ 1

Rule 2 Total probability: the sum of the probabilities of all possible outcomes is equal to 1 (or 1 0 0 % ).

The compliment of event 𝐴 , written as 𝐴  , refers to everything that is not   𝐴 .

Rule 3 𝑃 ( 𝐴 ) = 1 − 𝑃 ( 𝐴 ) 

Note: The compliment of event 𝐴 is sometimes also written as 𝐴 .

If two events 𝐴 and 𝐵 cannot occur at the same time, we say that they are mutually exclusive or disjoint events. In this case, the joint probability of 𝐴 and 𝐵 occurring at the same time is zero. We write this as 𝑃 ( 𝐴 ∩ 𝐵 ) = 0 . If events 𝐴 and 𝐵 are mutually exclusive, the probability that 𝐴 or 𝐵 occurs is the sum of their probabilities. That is,

Rule 4 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 )

If events 𝐴 and 𝐵 are not mutually exclusive, the probability that either 𝐴 or 𝐵 or both occur is

Rule 5 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) − 𝑃 ( 𝐴 ∩ 𝐵 )

Let us look at an example of compound probabilities for nonmutually exclusive events.

Example 2: Compound Probability: Probabilities for Nonmutually Exclusive Events

In a sample of 55 people, 28 have brown hair and 22 have blue eyes. 5 of them have neither brown hair nor blue eyes. What is the probability that a random person from the sample has at least one of these features?

Our compound events here are “brown hair”, and “blue eyes.” To find the probability that a person chosen at random from the sample of 55 people has at least one of these features, we can simply note that since 5 of the 55 have neither feature, all the rest must have at least one, that is, that 5 5 − 5 = 5 0 of the 55 have either brown hair or blue eyes or both.

The probability that a person chosen at random has either brown hair, blue eyes, or both is, therefore, 𝑃 ( ∪ ) = 5 0 5 5 = 1 0 1 1 . B r o w n h a i r B l u e e y e s

In essence, we have used our total probability rule to calculate this. Remembering that, for event 𝐴 , 𝑃 ( 𝐴 ) = 1 − 𝑃 ( 𝐴 )  and letting “ b r o w n h a i r ” B r = and “ b l u e e y e s ” B l = , then B r B l ∪  means neither brown hair nor blue eyes. And 𝑃 ( ∪ ) = 1 − 𝑃  ∪  = 1 − 5 5 5 = 5 0 5 5 = 1 0 1 1 . B r B l B r B l 

Hence, as noted, the probability that a person chosen at random from the sample has at least one of the features “brown hair” and “blue eyes”, is 1 0 1 1 .

In our next example, we will calculate compound probabilities for the results of two spinners.

Example 3: Compound Probability for Two Spinners

If these two spinners are spun, what is the probability that the sum of the numbers the arrows land on is a multiple of 5?

To find the probability that the sum of the numbers the two arrows land on is a multiple of 5, let us first put all possible outcomes into a table.

All possibilities for spinner 2 are in the blue block across the top and those for spinner 1 are in the red block down the left-hand side. The sums of the values on the two spinners are in the body of the table. So, for example, if spinner 2 lands with the pointer on 8 and spinner 1 lands with the pointer on 7, the sum is 15, which is below the column headed “8” and across the row labeled “7.”

To find the probability that the sum of the numbers on the two spinners is a multiple of 5, we need to know two things:

• How many of the results are multiples of 5,
• How many possibilities there are in total.

First, considering multiples of 5, we note that the range of values in the table is from 3 to 18. Between these two, the multiples of 5 are 5, 10, and 15. Let us mark the instances of 5, 10, and 15 in our table and count how many we have.

As we can see, there is one instance of the number 5 and two instances each of the numbers 10 and 15. We therefore have 1 + 2 + 2 = 5 “favorable outcomes,” that is, 5 instances where the sum on the two spinners is a multiple of 5.

We have not quite finished yet, however. Remember, we are working out the probability that the sum of the two numbers is a multiple of 5. The second number we need in order to calculate this is the number of possibilities in total. We can find this by multiplying the number of rows by the number of columns in the body of our spinner results table (the number of rows corresponds to the number of segments on spinner 1 and the number of columns is the number of segments on spinner 2).

The total number of possible outcomes is therefore 3 × 7 = 2 1 ; hence, 𝑃 ( 5 ) = 5 = 5 2 1 . s u m i s a m u l t i p l e o f n u m b e r o f w a y s s u m i s a m u l t i p l e o f t o t a l p o s s i b l e o u t c o m e s

The probability that the sum on the two spinners is a multiple of 5 is therefore 5 2 1 .

When we throw a coin, what happens on our second throw is not affected by what happened on our first throw. Whether we threw a heads or a tails on the first throw makes no difference to the probability of the outcome of the second throw, whereas when we take a ball from a bag, there is one less ball in the bag for our second selection. This affects the probability of the outcome of our second selection. The events “first ball red“ and “second ball red” are therefore not independent events.

Let us define “independence” of events before we look at some examples.

Definition: Independent and Dependent Events

Two events 𝐴 and 𝐵 are

• independent if the fact that 𝐴 occurs does not affect the probability of 𝐵 occurring,
• dependent if the fact that 𝐴 occurs does affect the probability of 𝐵 occurring.

For independent events, 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐴 ) × 𝑃 ( 𝐵 ) .

Our next example is a straightforward example of calculating probability for independent events.

Example 4: Calculating Probability for Independent Events

𝐴 and 𝐵 are independent events, where 𝑃 ( 𝐴 ) = 1 3 and 𝑃 ( 𝐵 ) = 2 5 . What is the probability that events 𝐴 and 𝐵 both occur?

Given that events 𝐴 and 𝐵 are independent, the probability that they both occur is 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐴 ) × 𝑃 ( 𝐵 ) = 1 3 × 2 5 = 2 1 5 .

Our next example shows how to calculate probabilities for selection without replacement, that is, for dependent events.

Example 5: Compound Probability: Taking Two Balls from a Bag without Replacement

A bag contains 22 red balls and 9 green balls. One red ball is removed from the bag and then a ball is drawn at random. Find the probability that the drawn ball is red.

To find the probability of drawing a second red ball from the bag, we note first that since there are 22 red balls and 9 green ones, there are 2 2 + 9 = 3 1 balls in total. The probability of drawing a red ball from the bag on our first pick is, therefore, 𝑃 ( ) = = 2 2 3 1 . R e d n u m b e r o f r e d b a l l s t o t a l n u m b e r o f b a l l s

Since we are keeping the first ball out of the bag, there is one less ball in the bag in total, so there are now 30 balls in the bag. And of those 30, there is one less red since the ball we took out was red. Hence, the number of red balls is now 21.

To help us work out the probability of drawing a second red ball from the bag, we can illustrate the problem using a tree diagram.

The probability of taking a second red ball having not replaced the first is found by multiplying the probabilities on the branches corresponding to “first ball red” and “second ball red”: 𝑃 ( ∩ ) = × = 2 2 3 1 × 2 1 3 0 = 2 2 × 2 1 3 1 × 3 0 = 4 6 2 9 3 0 = 7 7 1 5 5 ≈ 0 . 4 9 7 . R e d R e d r e d b a l l s i n i t i a l l y t o t a l b a l l s i n i t i a l l y r e d b a l l s l e f t t o t a l b a l l s l e f t

The probability that the second ball drawn is red is therefore 0.497. We can say that there is approximately a 5 0 % chance of choosing two consecutive red balls (since 0 . 4 9 7 × 1 0 0 % = 4 9 . 7 % ).

Note that we have actually used the formula for compound dependent events: 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐴 ∣ 𝐵 ) × 𝑃 ( 𝐵 ) . The probability 2 1 3 0 is the conditional probability of selecting a red ball given that a red ball has already been taken from the bag: 𝑃 ( 2 ∣ 1 ) R R . And 2 2 3 1 is the probability that the first ball selected is red. Hence, using the formula, we have 𝑃 ( 1 ∩ 2 ) = 𝑃 ( 2 ∣ 1 ) × 𝑃 ( 1 ) = 2 1 3 0 × 2 2 3 1 ≈ 0 . 4 9 7 . R R R R R

Compound probability is concerned with finding the probability of more than one event occurring.

When calculating probabilities for compound events, there are various possibilities to consider.

• Events may be mutually exclusive, in which case they cannot occur together. That is, there is no overlap between them. In this case, for events 𝐴 and 𝐵 , 𝑃 ( 𝐴 ∩ 𝐵 ) = 0 , so the probability of 𝐴 or 𝐵 occurring is given by 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) × 𝑃 ( 𝐵 ) . If events 𝐴 and 𝐵 are not mutually exclusive, then 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) − 𝑃 ( 𝐴 ∩ 𝐵 ) .

If the number of possible outcomes is small, it can be helpful to illustrate the events and their probabilities in a tree diagram. If there are too many outcomes for a tree diagram to be useful, it may be appropriate to illustrate all the possible outcomes in a table.

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Practice problems for probability of compound events: video lesson

In this video lesson, I solve several probability problems that involve compound events by listing all the possible outcomes or by drawing a tree diagram. Either way, we have the complete sample space, and we can figure out the probabilities just by writing the ratio of the favorable outcomes to all the possible outcomes (definition of simple probability).

The situations include:

• choosing two students from among five to clean the classroom
• tossing a coin 2 times (there are four possible outcomes)
• tossing a coin 3 times (there are eight possible outcomes)
• picking two cards randomly from a set of cards.

This lesson suits grades 7-8, and meets the Common Core standard 7.SP.8a and 7.SP.8b:

a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

Math Mammoth Grade 7 curriculum

Introduction to probability - video lesson

Probability of compound events - video lesson

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Probabilities of Compound Events

You can find the probability of compound events in the same way you find the probability of simple events: by writing it as a fraction with the number of favorable outcomes over the number of possible outcomes in the sample space. In this seventh-grade math worksheet, students are asked to find the probability of compound events using tables, tree diagrams, or organized lists to help them. With relatable real-world scenarios, this exercise is a great way to help students see how their math learning applies to everyday questions while also preparing them for higher-level probability topics down the road.

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Free Printable Probability of Compound Events worksheets

Math Probability of Compound Events: Discover a collection of free printable worksheets to help students master the concepts of compound events in probability. Ideal for teachers and learners alike.

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Probability of Compound Events worksheets are an essential tool for teachers who want to help their students master the concepts of probability in Math. These worksheets provide a variety of exercises and problems that focus on the calculation and interpretation of probabilities for compound events, which are events that involve the occurrence of two or more simple events. By incorporating Data and Graphing techniques, students can visualize the relationships between different events and better understand the underlying principles of probability. Teachers can use these worksheets to create engaging and interactive lessons that cater to different learning styles and help students develop critical thinking and problem-solving skills. With a wide range of Probability of Compound Events worksheets available, teachers can easily find the perfect resources to support their lesson plans and ensure their students' success in understanding this important mathematical concept.

In addition to Probability of Compound Events worksheets, teachers can also take advantage of Quizizz, an online platform that offers a variety of interactive quizzes and activities to help students practice and reinforce their understanding of Math, Data and Graphing, and Probability concepts. Quizizz allows teachers to create custom quizzes or choose from a vast library of pre-made quizzes, making it easy to find the perfect resources to complement their lesson plans. With Quizizz, students can work at their own pace and receive instant feedback on their progress, while teachers can monitor their students' performance and identify areas where additional support may be needed. By incorporating Quizizz into their teaching strategies, teachers can create a more engaging and effective learning environment that supports the development of essential mathematical skills and helps students excel in their understanding of Probability of Compound Events and other key concepts.

Compound Probability

Compound probability is the probability of two or more independent events occurring together. Compound probability can be calculated for two types of compound events, namely, mutually exclusive and mutually inclusive compound events. The formulas to calculate the compound probability for both types of events are different.

Compound probability is a concept that is widely used in the finance industry to assess risks and assign premiums to various policies. In this article, we will learn more about compound probability, its formulas, how to determine it as well as see various associated examples.

What is Compound Probability?

The compound probability of compound events (mutually inclusive or mutually exclusive) can be defined as the likelihood of occurrence of two or more independent events together. An independent event is one whose outcome is not affected by the outcome of other events. A mutually inclusive event is a situation where one event cannot occur with the other while a mutually exclusive event is when both events cannot take place at the same time. The compound probability will always lie between 0 and 1.

Compound Probability Formulas

There are two formulas to calculate the compound probability depending on the type of events that occur. In general, to find the compound probability, the probability of the first event is multiplied by the probability of the second event and so on. The compound probability formulas are given below:

Mutually Exclusive Events Compound Probability

• P(A or B) = P(A) + P(B)

Using set theory this formula is given as,

• P(A ∪ B) = P(A) + P(B)

Mutually Inclusive Events Compound Probability

• P(A or B) = P(A) + P(B) - P(A and B)
• P(A ∪ B) = P(A) + P(B) - P(A ⋂ B)

where A and B are two independent events, and P(A and B) = P(A) x P(B)

Compound Probability Example

Suppose a coin is tossed. The outcome of getting heads will be a simple event with a probability of 1 / 2. However, if the coin is tossed twice then the outcome of getting two heads will be a compound event. The compound probability of this event can be calculated as (1 / 2) x (1 / 2) = 1 / 4 or 0.25. This is an example of compound probability.

How to Find Compound Probability?

The steps to apply the compound formulas can be understood with the help of an example. Suppose the probability of Ryan failing an exam is 0.3 and the probability of Berta failing is 0.2. Then to find the compound probability of Ryan or Berta failing, the steps are as follows:

• Determine if the compound event is mutually exclusive or inclusive. This is an example of a mutually inclusive event.
• List the given probabilities. P(R) = 0.3 and P(B) = 0.2.
• Determine the correct compound probability formula. This is P(A or B) = P(A) + P(B) - P(A and B) for the given example
• Find P(A and B) which is given by P(A) x P(B). Thus, 0.2 x 0.3 = 0.6.
• Plug the values into the formula to get the result. Thus, the compound probability for the example is 0.44

Related Articles:

• Probability Rules
• Events in Probability
• Exhaustive Events
• Dependent Events
• Mathematical Induction

Important Notes on Compound Probability

• Compound probability is the likelihood of occurrence of two independent compound events together.
• Compound probability can be calculated for mutually exclusive and mutually inclusive compound events.
• P(A or B) = P(A) + P(B) and P(A or B) = P(A) + P(B) - P(A and B) are the compound probability formulas.

Examples on Compound Probability

Example 1: There are 40 girls and 30 boys in a class. 10 girls and 20 boys like tennis while the rest like swimming. If a student is selected at random then what is the probability that it will be a boy or a girl.

Solution: If a student is selected it can only be a girl or a boy. Thus, the probability that the selected student will be a girl or a boy is 1.

Answer: P(Boy or Girl) = 1

Example 2: If a dice is rolled then find the compound probability that either a 2 or 3 will be obtained.

Solution: P(2) = 1 / 6

P(3) = 1 / 6

As this is an example of a mutually exclusive event thus, the compound probability formula used is

P(2 or 3) = (1 / 6) + (1 / 6)

Answer: P(2 or 3) = 1 / 3

Example 3: Find the compound probability of selecting 5 or a black card from a deck.

Solution: Total number of cards in a deck = 52

Number of cards in each suit = 13

P(Black card) = 26 / 52

Total number of 5s in a deck = 4

P(5) = 4 / 52

Number of 5s in black cards = 2

P(Black card and 5) = 2 / 52

This is a mutually inclusive event thus, the compound probability formula is

P(Black card or 5) = P(Black card) + P(5) - P(Black card and 5)

Answer: P(Black card or 5) = 7 / 13

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FAQs on Compound Probability

What is the meaning of compound probability.

Compound probability, in probability and statistics , is the probability that describes the chance that two or more independent events will occur together. It is determined by multiplying the probabilities of the occurring events.

What is the Formula for Compound Probability?

There are two formulas available for calculating the compound probability. These are given as follows:

What is the Formula for Compound Probability in Set Theory?

The compound theory formulas expressed using set operations are given as follows:

What are the Types of Events Used in Compound Probability?

There are two types of events used in compound probability. These are mutually exclusive and mutually inclusive compound events.

Can Compound Probability be Greater Than 1?

The compound probability value will always lie between 0 and 1. 0 indicates that the event will never occur and 1 denotes that the event will definitely take place.

What is the Difference between Simple and Compound Probability?

Simple probability is used to give the likelihood that one event will take place. On the other hand, compound probability gives the probability of occurrence of more than one separate event.

How to Calculate Compound Probability?

The steps to calculate the compound probability are as follows:

• Determine if the event is mutually inclusive or mutually exclusive.
• Apply the corresponding formula.

Module 5: Probabilities of Compound Events

Home       Student Resources       Teacher Resources

Probability is widely used in many career fields to help researchers and companies make reasonable predictions based on data collected.  Thus, it is important to understand how to calculate and evaluate probabilities for simple and compound events. 

In this module, you will:

• Build on your prior understanding of probability to include compound events. 
• Examine independent and dependent events, and describe events as subsets of a sample space. 
• Determine whether events are independent or dependent.
• Apply the Addition Rule and interpret answers in terms of the context provided. 
• Construct and interpret two-way tables of data.

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Finding probabilities of compound events.

Cecil has two six-sided dice, a red one and a white one.

If Cecil throws the two dice, what is the probability that the red die is a 1? What is the probability that the sum of the dice is 7?

• Are the two events described part (a) independent? Explain.
• What is the probability that the red die is a 2? What is the probability that the sum of the two dice is 10?
• Are the two events described in part (c) independent? Explain.

IM Commentary

The goal of this task is to study probabilities of compound events. The events described are at the level of the seventh grade standards. This is a deliberate choice so that students can focus on the meaning of independence in probability. Once these calculations have been made, the teacher may wish to explore some more complex scenarios with more dice or with a different context.

The standard S-CP.A.3 also discusses the mathematical meaning of probabilistic independence: events $A$ and $B$ are independent when

In the table below the first number denotes the outcome of the red die and the second number the outcome of the white die so $(5,3)$ means that the red die is a 5 and the white die is a 3.

The cases where the first die is a 1 or where the sum is 7 have been highlighted. From the table we can see that the probability that the red die is a 1 is $\frac{1}{6}$ and the probability that the sum of the two dice is 7 is also $\frac{1}{6}$.

These two events are independent. To understand why we need more information, however, than we found in (a). If the red die had been a 2, the probability that the sum of the two is 7 would still be $\frac{1}{6}$, the one successful outcome happening when the white die is a 5. More generally, the probability that the sum is a 7 with any specified value of the red die is the same as the probability that the sum is a 7 with no specified value.

Conversely, the red die is 1 in exactly 1 out the 6 ways of getting 7. So if we ask for the sum of the two dice to be 7, the probability for the red die being a 1 is still $\frac{1}{6}$, the same as it is with no restriction on the sum of the dice. From this and the previous paragraph, we conclude that the two events described in (a) are independent.

In the table below the cases where the red die is a 2 or where the sum of the two dice is 10 have been highlighted.

The probability that the first die is a 2 is $\frac{1}{6}$ while the probability that the sum of the two is 10 is $\frac{1}{12}$.

As in part (b) we must investigate further to determine if the events are independent. Notice that the probability that the sum of the two dice is 10, provided that the red die is a 2, is 0. Similarly if the first die is a 1, 2, or 3 the probability that the sum of the two dice is 10 is 0. When the first die is a 4, 5, or 6, however, the probability that the sum is 10 is $\frac{1}{6}$. In none of these cases is the probability equal to the probability that the sum of the dice is 10, namely $\frac{1}{12}$. So the first die being 2 and the sum being 10 are not independent events.