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2. Probabilities with Multiple Events
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Chapter 12
2. 

Probabilities with Multiple Events

Understanding probabilities with multiple events is essential for analyzing situations involving uncertainty. This lesson introduces sample spaces and tree diagrams to visualize possible outcomes. It also explores compound events, which involve multiple conditions or scenarios, and demonstrates how simulations can be used to model and analyze probabilities in real-world contexts. By mastering these concepts, students can solve complex probability problems and apply these skills to fields like statistics, science, and decision-making.
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Probabilities with Multiple Events
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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.

Deli compressed.JPG

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

Bread compressed.jpg

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. rcl White & → & W Rye & → & R Similarly, the first letter of each type of deli meat can be used to represent the meat on the sandwich. rcl Roast Beef & → & R Ham & → & H Turkey & → & 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.

Cheese shop compressed.jpg

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.

Making the tree diagram. The first row from left to right: Bread, Meat, Cheese, Sample Space. The bread column lists White and Rye.

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.

Making the tree diagram. The first row from left to right: Bread, Meat, Cheese, Sample Space. The bread column lists White and Rye. The meat columns lists Roast Beef, Ham, Turkey twice, for each bread.

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

Making the tree diagram. The first row from left to right: Bread, Meat, Cheese, Sample Space. The bread column lists White and Rye. The meat column lists Roast Beef, Ham, Turkey twice, for each kind of bread. The cheese column lists American, Swiss, Provolone three times, for each kind of meat.

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.

Tree diagram

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 * 3 * 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. A coin is flipped_(first event) 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 * 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 a3) = 1/12
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.

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

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.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swiss, and provolone cheese. The white bread option is highlighted.

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

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swiss, and provolone cheese. Highlight on white bread followed by roast beef and turkey options.

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

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swiss, and provolone cheese. Highlight on white bread followed by roast beef provolone, and turkey provolone.
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=2/18 Notice that this fraction can be simplified.
2/18
Rewrite

Rewrite 18 as 2 * 9

2/2* 9
CancelCommonFac

Cancel out common factors

2/2* 9
SimpQuot

Simplify quotient

1/9
This means that the probability that Magdalena's mom will like a sandwich made at random with the given ingredients is 19.
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)=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.

Crane game compressed.jpg

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:& 9 Failed Attempt:& 0, 1, 2, 3, 4, 5, 6, 7, or8 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. ccccc 50566 & 17559 & 77876 & 50710 52189 & 65203 & 11852 & 08484 22207 & 53733 & 73619 & 08260 53334 & 82347 & 61241 & 00428 84909 & 31293 & 80378 & 22341 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. cccc 50566 & 17559 & 77876 & 50710 52189 & 65203 & 11852 & 08484 22207 & 53733 & 73619 & 08260 53334 & 82347 & 61241 & 00428 84909 & 31293 & 80378 & 22341 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) = 5/20 Notice that this fraction can be simplified.
5/20
Rewrite

Rewrite 20 as 5 * 4

5/5 * 4
CancelCommonFac

Cancel out common factors

5/5 * 4
SimpQuot

Simplify quotient

1/4
The simulation showed that the experimental probability of getting at least one plush doll from the crane game is 14. This probability can be written as a percentage. 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. 2coffee types * 5 milk options * 4sweeteners = 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. 1/40 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! ... Or maybe she just got lucky.
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Probabilities with Multiple Events
Exercise 3.1

Magdalena and her sister are close to finishing a board game. Magdalena needs to roll a 5 or higher to win, or else her sister will win. She has two options available — she can either roll a six-sided die or she can roll two four-sided dice. If she rolls the two four-sided dice, the results are added together.

Which option gives Magdalena better odds at winning?
Suppose Magdalena is using her best option. Who is more likely to win?

Magdalena needs to roll a 5 or higher to win. Let's consider both of Magdalena's options. The first option is to roll a six-sided die. Let's define the sample space of the experiment of rolling the six-sided die. { 1 , 2 , 3 , 4 , 5 , 6} If Magdalena rolls a 5 or 6, she wins. These are 2 out of the 6 possible outcomes. Let's divide 2 by 6 to find the probability that Magdalena wins using one six-sided die. 2/6 Since we are about to compare it to another probability, it is better if we write it as a percent. Let's do it!

Now let's consider her second option — rolling two four-sided dice. Since this is a compound event, we will use a tree diagram to find the sample space of the experiment. We will also need to find the sums of the outcomes of both dice.

The sample space consists of 16 elements. Let's now calculate the sums of the outcomes!

From here, we identify which sums are 5 or greater.

There are 10 outcomes that give a sum of 5 or more. We can find the probability of the compound event by dividing the number of favorable outcomes, 10, by the number of possible outcomes, 16. 10/16 Now let's write this probability as a percent so we can compare it to the one before.

We found that Magdalena has a 33.3 % chance of winning with one six-side die and a 62.5 % chance of winning using two four-sided dice. This means that Magdalena has greater odds of winning using the two four-sided dice.


We found in Part A that Magdalena has a 62.5 % chance of winning if she chooses to use two four-sided dice for her final roll. We can find the probability that her sister still wins the game by subtracting this probability from 100 %. 100 % - 62.5 % = 37.5 % Since Magdalena has a 62.5 % chance of winning, her sister only only has a 37.5 % chance of winning. This means that Magdalena is more likely to win in this scenario!

E
Hint & Answer
S
Solution

Emily is looking forward to getting a particular prize from a gift box in her mobile game. The game rolls a random number between 0 and 9, and Emily will get the prize she wants if the roll is a 9. Each gift box costs $1 and she has $5 to spend on this game.

Prize box with a random number being chosen above it.
Before deciding to spend any money, Emily goes to her laptop and finds an online simulator. She has $5 to spend, so the simulator will generate five random numbers. Emily runs the simulator 20 times and gets the following results. cccc 42560 & 42250 & 64652 & 67079 14164 & 40552 & 22814 & 30141 72471 & 38141 & 66609 & 52222 08899 & 71423 & 90959 & 07830 00815 & 92032 & 58928 & 39420 Find the experimental probability of Emily getting more than one of the prize she wants.

Emily wants to determine how likely it is that she will win more than one of a particular prize if she spends $5. After running the simulation, she got the following numbers. cccc 42560 & 42250 & 64652 & 67079 14164 & 40552 & 22814 & 30141 72471 & 38141 & 66609 & 52222 08899 & 71423 & 90959 & 07830 00815 & 92032 & 58928 & 39420 Since the event of getting the prize is represented by the number 9, we will look for the trials that have more than one 9. cccc 42560 & 42250 & 64652 & 67079 14164 & 40552 & 22814 & 30141 72471 & 38141 & 66609 & 52222 08899 & 71423 & 90959 & 07830 00815 & 92032 & 58928 & 39420 Only 2 out of the 20 trials made in the simulation show the outcome of getting more than one of the desired prize from the gift boxes. This lets us find the experimental probability of winning this prize more than once when spending $5 on the gift boxes. 2/20 Let's simplify this fraction!

The simulation suggests that Emily will get more than one of the prize she wants one in ten times that she spends $ 5 on the gift boxes. Note that this experimental probability is given by this particular experiment. Keep in mind that different simulations will give different numbers, which might result in a different experimental probability.

E
Hint & Answer
S
Solution

Probabilities with Multiple Events

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