2. Probabilities with Multiple Events
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Chapter 12
2.
Probabilities with Multiple Events
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Why
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.
Lesson Settings & Tools
| | 11 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Probabilities with Multiple Events
Slide of 11
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.
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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 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.
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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} .
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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?
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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!
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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.
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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.
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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 * 3 * 3 = 18
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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
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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 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.
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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=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.
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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.
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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:& 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?
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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!
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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.
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Probabilities with Multiple Events
Exercise 2.1
Suppose a coin is flipped three times in a row. What is the probability of getting tails at least once?
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We want to find the probability of getting tails at least once when flipping a coin three times in a row. We need to know the sample space of the experiment before we can find this probability. Let's start by drawing a tree diagram to help us find the sample space. The first step to creating the tree diagram is writing the different outcomes of the first throw.
We now write the possible outcomes of the second throw next to the outcomes of the previous throw. We will use segments to connect these throws.
Then we do the same for the third throw.
As the last step of the tree diagram, we define the sample space of the experiment.
The sample space consists of eight elements. Getting at least one tails means that we can either have one tails, two tails, or three tails. Let's highlight these outcomes from our tree diagram.
Notice that we have seven different outcomes that showed tails at least once. We can find the probability of this event by dividing 7 by the total number of possible outcomes, 8. 7/8
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Two six-sided dice are rolled. What is the probability that the numbers rolled add up to 11?
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We are told that two six-sided dice are rolled. We want to find the probability that the numbers rolled add up to 11. We first need to know how many elements there are in the sample space. Because there are a lot of outcomes, making a tree diagram would require a lot of space. Instead, we can multiply the number of possible outcomes of each rolled die together. 6 * 6 = 36 The given experiment has 36 different outcomes. Since we are interested in the sum of the rolls, we need to consider the ways in which we can add the results of the rolls and get 11. This can only happen when we roll a combination of the numbers 5 and 6. 5+6 &= 11 6+5 &= 11 Notice that if either roll is less than 5, it is not possible to get a sum of 11. This means that there are only 2 favorable outcomes. Knowing this, we can find the probability of rolling an 11 on two six-sided dice by dividing the number of favorable outcomes by the total number of possible outcomes. 2/36 Let's simplify the fraction.
The probability of getting the dice to add up to exactly 11 is 118.
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Magdalena's cat is pregnant. The vet says that she is expecting triplets! Suppose that each kitten has an equal chance of being born male or female. What are the odds that Magdalena's cat gives birth to at least one male and one female kitten?
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We are told that Magdalena's cat is about to give birth to three kittens. To find the probability that she gives birth to at least one male and a female kitten, we will first define the sample space of this experiment. Let's make a tree diagram! The first step to making a tree diagram is to list the possible outcomes of the sexes of the firstborn kitten.
We now list the possible sexes of the second kitten and use segments to connect to their littermate.
The third birth is listed in a similar way.
We finish the diagram by listing each element of the sample space.
Now let's look at the sample space and find the outcomes that contain at least one male and one female kitten. We will highlight these outcomes.
We can see that in 6 out of the 8 total possible outcomes, at least one male and one female kitten were born. The quotient of these numbers gives us the probability of this compound event. 6/8 Notice that we can simplify this fraction. Let's do it!
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Magdalena asked her sister if she could borrow her laptop while her sister was out at the mall with her friends. Her sister said that it was fine, but she forgot to tell Magdalena the password to unlock the laptop. If the password is a four-digit number, what are the odds of getting the password right on the first try?
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We are told that the computer password is made up of four digits. Digits are the numbers from 0 to 9. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 This means that there are a total of 10 options for the first digit, 10 options for the second, and so on until the password is complete. To find the total number of possible passwords, we need to multiply 10 by itself four times. 10 * 10 * 10 * 10 = 10 000 There are 10 000 possible four-digit passwords. Only 1 of them is the correct one. We can find the probability of guessing the correct password on the first try by dividing these numbers. 1/10 000 The probability of guessing the password correctly on the first try is 110 000. Magdalena should probably call her sister rather than try to guess the password.
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Probabilities with Multiple Events
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