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Events are typically not isolated — they often occur alongside other events. These types of events are called compound events and represent the odds of two or more events happening together. Compound events are used to describe a wider variety of scenarios using the basics of probability. This lesson will discuss compound events and provide some tools that are useful when studying them.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Getting the Right Coffee

One Sunday morning, Magdalena went grocery shopping with her parents. On their way home, the family stopped at a local coffee shop to get some drinks. As they waited at the ordering screen, Magdalena got a text from her sister, asking her for a coffee. Unfortunately, she did not elaborate on how she would like her coffee and now she will not respond to calls or texts.

Magdalena's sister asking for a coffee in a text message

Magdalena knows her sister usually gets an Americano, but she does not know if she likes it black or with milk or sweetener. Since they are already in line and there are people waiting behind them, there is no time to waste! Magdalena will have to guess how her sister would like her coffee.

Ordering screen applet asking for the details about the american coffee Magdalena is about to order

There are two options for the coffee, normal and decaf. It is also possible to add whole, skim, soy, or almond milk (or no milk at all). Finally, the beverage can be sweetened with regular sugar, raw sugar, or stevia, or it can come unsweetened. What are the chances that Magdalena randomly orders the coffee exactly the way her sister likes it?

Discussion

Sample Space

To determine the probability of Magdalena guessing her sister's order correctly, she first needs to know the different combinations of coffee, milk, and sweetener that she can order. The set of all these possible combinations is known as the sample space.

The sample space of an experiment is the set of all possible outcomes. For example, when flipping a coin, there are two possible outcomes: heads, H, or tails, T. The sample space of flipping a coin is ${H, T} .$

Coin flip outcomes: Head and Tail in the sample space.

Example

Picking the Ingredients for a Sandwich

Magdalena and her family have very different tastes in sandwiches. For example, Magdalena likes a simple ham sandwich made with white bread, while her dad likes roast beef sandwiches on rye. Because of this, they usually buy three different types of deli meat: roast beef, ham, and turkey.

On the bread side, everyone likes white bread aside from her dad. Magdalena wonders how many different sandwiches can be made from these ingredients.

Define the sample space of all the possible sandwiches that can be made using white bread, rye bread, roast beef, ham, and turkey. Assume that only one type of bread and one type of deli meat can be used on each sandwich. How many different sandwiches can be made?

Hint

Use different combinations of letters to represent each type of sandwich. For example, let WR be a roast beef sandwich made with white bread. Count how many different combinations can be made.

Solution

There are two different bread options for each sandwich, white and rye bread. The first letter of each type of bread can be used to represent which one will be used in a sandwich.

White Rye \to \to W R

Similarly, the first letter of each type of deli meat can be used to represent the meat on the sandwich.

Roast Beef Ham Turkey \to \to \to R H T

Notice that the letter R is used both for rye bread and for roast beef. This is not a problem at all as long as it is each sandwich option is ordered the same. Here, the first letter corresponds to the bread and the second letter indicates the deli meat. Knowing this, the sample space can now be defined.

{WR, WH, WT, RR, RH, RT}

Now that the sample space has been defined, the number of elements can be counted. There are a total of six different sandwiches that can be made from the given options of bread and deli meat!

Discussion

Tree Diagrams

A tree diagram is an illustration of all the possible combinations that can be made from a given set of objects. Tree diagrams are useful for defining the sample space of an experiment when multiple choices are available. As an example, consider the different pasta dishes that can be made with different types of pasta, sauce, and cheese.

Tree diagram

In this situation, there are two different types of pastas, three different sauces, and two different cheeses. The tree diagram describes the sample space that contains all

12

different dishes that can be made by combining the given ingredients.

Example

Adding Cheese to the Sandwiches

Magdalena's family also likes different types of cheese. From the wide variety of cheeses available at the store, they typically buy American, Swiss, and provolone.

Magdalena now wonders how many different sandwiches her family can make using the different types of bread, deli meat, and cheese. Assume that only one type of meat and cheese can be used per sandwich. Use a tree diagram to find how many different sandwiches can be made from the the breads, meats, and cheeses that the family usually buys.

Hint

There are three categories to consider when making the sandwich: bread, meat, and cheese. When moving to the next category, all the options need to be considered.

Solution

Adding cheese to the sandwiches makes the task of finding the sample space a little harder. A tree diagram can provide a useful visual aid to help find how many different combinations of bread, meat, and cheese can be made from the given options. Start by listing the two bread options.

Next, list the three options for the deli meat next to each bread option. Use segments to connect the bread types to each of the meat options available.

Similarly, list the three cheese options next to each meat option. Draw segments connecting meat options to cheese options.

Each possible outcome is represented by a path in the tree diagram. Use the first letter of each ingredient to represent all the possible sandwiches that can be made with the given ingredients. This process completes the definition of the sample space.

The sample space consists of

18

elements, which means that

18

different sandwiches can be made with the given ingredients. Notice that the same result can be found by multiplying the number of options in each category. There are

2

different types of bread,

3

different deli meats, and

3

different cheeses.

2 \cdot 3 \cdot 3 = 18

Discussion

Compound Event

A compound event is an event that is a combination of two or more separate events. Below is an example.

first event A coin is flipped and a die is rolled second event .

The number of elements of the sample space of the compound event can be found by multiplying the number of possible outcomes of each individual event.

Event	Number of Outcomes
Flipping a coin	$2$
Rolling a die	$6$
Flipping a coin and rolling a die	$2 \times 6 = 12$

One possible outcome of this compound event is getting tails on the coin and rolling a $3$ on the die. To find the probability of this outcome, divide this $one$ possibility by the number of possible outcomes of the compound event.

P (Flipping a tails and rolling a 3) = \frac{1}{1 2}

Example

Will Mom Like the Sandwich?

Magdalena is making lunch for the family. She knows that her mom loves provolone cheese — she often eats some slices with salami and olives as a snack! Magdalena asked her mother what her favorite type of meat for sandwiches is.

Magdalena having a conversation with her mom about how she likes her sandwiches. Magdalena's mom says she just does not like ham.

Magdalena remembers that her father is the only one in the family who likes rye bread. She wonders if she could create a sandwich her mother would like if she randomly chose the bread, meat, and cheese. Find the probability that Magdalena's mom likes a sandwich made with random ingredients.

Hint

Use a tree diagram showing all the sandwiches that can be made with the rye and white breads, provolone, Swiss, and American cheeses, and roast beef, ham, and turkey. Divide the number of favorable outcomes by the total number of possible outcomes.

Solution

Magdalena is making sandwiches for her family. Choosing the bread, meat, and cheese are all separate events, with the entire process of making a sandwich being the compound event made up of the simple events. A tree diagram comes in handy for visualizing all of the possible outcomes of this compound event.

To find the probability that Magdalena's mom likes a sandwich made from random ingredients, the number of favorable outcomes needs to be divided by the total number of outcomes. Her mom prefers white bread.

The next choice is the meat. Magdalena's mom likes roast beef and turkey, so highlight these options in the tree diagram.

Finally, continue along the branches to identify which ones lead to provolone cheese.

There are

2

favorable outcomes out of the

18

possible outcomes. To find the probability of making a sandwich that Magdalena's mom will enjoy, divide the number of favorable outcomes by the total number of possible outcomes.

Probability = \frac{2}{1 8}

Notice that this fraction can be simplified.

\frac{2}{1 8}

Rewrite $18$ as $2 \cdot 9$

\frac{2}{2 \cdot 9}

CancelCommonFac

Cancel out common factors

\frac{2}{2 \cdot 9}

Simplify quotient

\frac{1}{9}

This means that the probability that Magdalena's mom will like a sandwich made at random with the given ingredients is

\frac{1}{9} .

Discussion

Simulation

A simulation is a model that imitates a real-life process or situation. Simulations are often used as probability models to make predictions about real-life events. Consequently, the experimental probability of an event occurring can be estimated by simulating the event.

P (event) = \frac{Number of times event occurs}{Number of trials}

In general, simulations are used when actual trials of some experiment are impossible or unreasonable to conduct.

Example

The Crane Game

When her family was leaving the grocery store, Magdalena saw a crane game filled with cute plush dolls.

It costs fifty cents to play the crane game, and Magdalena had $$ 2.50,$ so she asked her dad if she could play. He told her that the odds of winning a plush doll are about $1$ in $10,$ so maybe it was not worth it to play the game.

After she got home, Magdalena thought of making a crane game simulator mobile game. She coded each attempt at the crane game as the generation of a random number between

0

and

9 .

Since the odds of winning are one out of ten, one out of the ten numbers represents a success.

Successful Attempt : Failed Attempt : 9 0, 1, 2, 3, 4, 5, 6, 7, or 8

Magdalena will test the prototype of the app with her previous predicament. Since she had

$ 2.50,

each trial of the simulation will consist of five random numbers. The simulator repeats the trial

20

times — imagine spending that much money on a crane game! She ran the simulation, which returned the following numbers.

5056652189222075333484909175596520353733823473129377876118527361961241803785071008484082600042822341

Using the data from the simulation, what is the experimental probability of getting at least one plush doll when spending

$ 2.50

to play the crane game?

Hint

Identify the outcomes of the simulation that contain at least one success. Then, divide the number of successes by the total number of trials.

Solution

Since a

9

represents winning a plush doll from the crane game, start by identifying the trials that contain at least one

9 .

5056652189222075333484909175596520353733823473129377876118527361961241803785071008484082600042822341

There were

5

successes out of the

20

simulated trials. To find the experimental probability of getting at least one plush doll from the crane game, divide the number of successes by the total number of trials.

P (getting at least one plush doll) = \frac{5}{2 0}

Notice that this fraction can be simplified.

\frac{5}{2 0}

Rewrite $20$ as $5 \cdot 4$

\frac{5}{5 \cdot 4}

CancelCommonFac

Cancel out common factors

\frac{5}{5 \cdot 4}

Simplify quotient

\frac{1}{4}

The simulation showed that the experimental probability of getting at least one plush doll from the crane game is

\frac{1}{4} .

This probability can be written as a percentage.

\frac{1}{4} = 25 %

A probability of

25 %

looks high enough for Magdalena. Maybe it would have been worth a shot to play back at the restaurant? However, keep in mind that each time the simulation is run, the experimental probability may change!

Closure

A Lucky Coffee

Earlier this morning, Magdalena found herself in the predicament of having people waiting in line behind her at the coffee shop as she tried to figure out what coffee her sister would like.

Ordering screen applet asking for the details about the americano coffee Magdalena is about to order

A tree diagram could be used to find how many different types of americano Magdalena could order but since there are so many options, the diagram would be huge. Instead, remember that the number of elements in the sample space of possible outcomes in a compound event can be found by multiplying the number of available options.

2 coffee types \cdot 5 milk options \cdot 4 sweeteners = 40

There are

40

ways to order an americano at the cafe. This means that Magdalena only has a

1

in

40

chance of getting her sister's coffee the exact way she likes it.

\frac{1}{4 0}

Magdalena decided to order it regular, no milk, with stevia. Amazingly, the order was exactly right! Magdalena must have unconsciously remembered how her sister ordered her coffee. What a great sister she is!

\dots

Or maybe she just got lucky.

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