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Events are typically not isolated, and they can occur along with other events. The concept of compound event is used to understand 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 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 particular Sunday morning, Magdalena went out with her parents to do the groceries. On their way back to home, they stopped at a drive-thru coffee shop to get some drinks. As they were on 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.

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

Magdalena knows her sister always asks for an American. However, she does not know if she likes it black, or if she puts any kind of milk in it. On top of that, she does not know whether to sweeten it or not! Since they are already in line, and there are people waiting behind them, Magdalena is short on time. Magdalena will have to make a guess on 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 either whole, skim, soy, or almond milk (or no milk at all). Finally, it can be sweetened with regular sugar, raw sugar, stevia, or unsweetened. What are the chances that Magdalena randomly orders the coffee exactly as her sister would have liked?

Discussion

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. Therefore, the sample space 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 a very diverse taste on sandwiches. For example, Magdalena likes a simple ham sandwich made with white bread. On the other hand, her dad likes a roast beef rye sandwich. Because of this, they usually buy three different types of deli meat: roast beef, ham, and turkey.

Deli compressed.JPG

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

Bread compressed.jpg

Write the sample space of all the possible sandwiches that can be made using white bread, rye bread, roast beef, ham, or turkey. Assume that only one type of deli meat can be used on each. 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 written.

Solution

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

White \to W Rye \to R

Similarly, the first letter of each type of deli meat can be used.

Roast Beef \to R Ham \to H Turkey \to 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 understood that the first letter corresponds to the bread, and the second letter to the deli meat. Knowing this, the sample space can now be written. Keep in mind that no deli meat is to be repeated.

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

Now that the sample space has been written, the number of elements can be counted. There are a total of

6

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 obtaining the sample space of an experiment when multiple choices are available. As an example, consider the different pasta recipes that can be made with different types of pasta, sauce, and cheese.

Tree diagram

In the above diagram, there are

2

different types of pastas,

3

different sauces, and

2

different cheeses. A tree diagram illustrates the sample space containing all

12

different options 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 existing cheese available, they typically buy American, Swizz, and provolone.

Cheese shop compressed.jpg

Magdalena now wonders how many different sandwiches can be made using the different types of bread, deli meat, and cheese. Assume that only one type of deli meat and cheese can be used. Use a tree diagram to find how many different sandwiches can be made from the above ingredients.

Hint

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

Solution

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

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

Similarly, next to each meat option list the three cheese options. 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 way, the sample space can be obtained.

The sample space consists of

18

elements. This 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 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	$2 \times 6 = 12$

In the above example, $one$ possible outcome of the compound event is flipping a tails and rolling a $3 .$ 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 knows that her mom loves provolone cheese. She even usually eats some slices of ham with olives as a snack! Magdalena now asks her mom what is her favorite type of meat.

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

Magdalena recalled that only her dad likes rye bread, so her mom should like white bread. Magdalena decided to use the tree diagram she made when exploring all the possible sandwiches.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swizz, and provolone cheese

What is the probability that Magdalena's mom likes a sandwich made with random ingredients?

Hint

Divide the number of favorable outcomes by the total number of outcomes.

Solution

Consider the given tree diagram.

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. Magdalena's mom prefers white bread.

From here, the meat can be either roast beef or turkey. Highlight these options in the tree diagram.

From both of these points, continue to the branches that contain provolone cheese.

There are

2

favorable outcomes out of the

18

possible outcomes. Find the probability by doing the division.

Probability = \frac{2}{1 8}

Notice that the above 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 from the given ingredients is

\frac{1}{9} .

Discussion

Simulation

A simulation is a model that imitates a real-life process or situation. In particular, a simulation can be used as a probability model to make predictions about real-life events. Consequently, the experimental probability can be estimated by simulating the events.

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

As Magdalena and her parents were leaving from the groceries store, Magdalena saw a crane game filled with cute plush dolls. Playing the crane game costs $$ 1,$ and Magdalena has $$ 5,$ so she asks her dad if she can play. Magdalena's dad says that the odds of winning a plush doll are about $1$ in $10,$ so he says that it is not worth it.

As soon as Magdalena is in her home, she goes to her laptop and finds an online simulator. In this case, each crane game attempt is simulated as a number between

0

and

9 .

Since the odds of winning are one out of ten, then one out of ten numbers will represent a success.

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

Since Magdalena had

$ 5,

five random numbers will be generated in succession. To obtain more relevant results, the experiment will be repeated

20

times. Imagine spending that much money on a crane game! Magdalena now runs the simulation and obtains the following numbers.

5056652189222075333484909175596520353733823473129377876118527361961241803785071008484082600042822341

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

$ 5

to play the crane game?

Hint

Identify which outcomes of the experiment contain at least one success. Divide the number of successes by the total number of trials.

Solution

Since a

9

represents getting a plush doll from the crane game, begin by identifying which trials contain at least one

9 .

5056652189222075333484909175596520353733823473129377876118527361961241803785071008484082600042822341

There were

5

successes out of the

20

trials that were simulated. Find the experimental probability of getting at least one plush doll from the crane game by dividing the number of successes by the number of trials.

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

Notice that the above 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 %

Keep in mind that by doing a different simulation different probabilities can be found.

Closure

A Lucky Coffee

Magdalena was previously found in the predicament of having people waiting on the line of the drive-thru coffee while not knowing exactly what her sister wanted.

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

A tree diagram can be used to find how many different orders are possible. Since there are plenty of options, the diagram will be huge. Instead, consider that there are

2

options for the type of coffee,

5

options for the type of milk, and

4

options for the sweetener. The number of elements in the sample space can be found by multiplying the number of available options.

2 \cdot 5 \cdot 4 = 40

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. It was a big surprise finding out that this is exactly how Magadelena's sister likes it! It is more likely that Magdalena unconsciously recalled how her sister orders her coffee. Anyways, we can say this was a very lucky coffee!

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