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| 11 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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.
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}.
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?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.
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.
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.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.
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.
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.
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.
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.
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.
Rewrite 18 as 2⋅9
Cancel out common factors
Simplify quotient
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.
Identify the outcomes of the simulation that contain at least one success. Then, divide the number of successes by the total number of trials.
Rewrite 20 as 5⋅4
Cancel out common factors
Simplify quotient
There are plenty of board games that use eight-sided dice to be played. The faces of this type of die range from 1 to 8.
Find the sample space of the possible outcomes of throwing an eight-sided die.We are told that an eight-sided die has eight faces. A number from 1 to 8 is written on each face, so there are eight possible outcomes. We can find the sample space of the possible outcomes of rolling the die by listing every possible number it could land on. {1, 2, 3, 4, 5, 6, 7, 8} Keep in mind that if the die is fair, all outcomes are equally likely to occur.
A local Italian restaurant likes to keep their menu simple.
They offer three different pizza toppings: pepperoni, salami, and anchovies. They also have three different sizes: small, medium, and large. Find the number of elements in the sample space that corresponds to the different one-topping pizzas that can be ordered in the restaurant.We are asked to find the number of elements in the sample space that corresponds to the different one-topping pizzas that can be ordered in the restaurant. A tree diagram can help us count these options. Let's begin by writing the different pizza toppings that the restaurant has to offer.
Next, we list all three options for the size of the pizza next to each topping. We use segments to connect the toppings and the sizes of the pizza.
We now can define the sample space. We will use the first letter of the topping and the first letter of the size to represent each of the possible outcomes of a customer ordering a pizza.
If we count the number of elements in the sample space column, we can see that there are a total of 9 different one-topping pizzas that can be ordered in the restaurant.
We are asked to find the probability of getting heads twice in a row when flipping a coin. Let's begin by finding the sample space of the experiment. We can use a tree diagram to help us find the sample space. We start the diagram by writing the different outcomes of the first throw.
Next, we list the outcomes of the second throw next to each outcome of the first throw. We use segments to connect both throws.
This lets us write the sample space of the experiment.
Since we are interested in the probability of getting heads twice in a row, we will first identify this outcome on the diagram.
Out of the four possible outcomes of the experiment, there is only one way to get heads twice. Finally, we can find the probability of getting this outcome by dividing the number of favorable outcomes, 1, by the total number of possible outcomes, 4.. 1/4 The probability of getting heads twice when flipping a coin twice is 14, or 25 %.
We are asked to find the probability of getting heads exactly twice when flipping a coin three times in a row. We will begin by defining the sample space of the experiment. We can use a tree diagram to help us find the sample space. We start creating the tree diagram by writing the different outcomes of the first throw.
Next, we 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 8 elements. Notice that we have three different outcomes where we get heads exactly twice.
Finally, we are ready to find the probability of getting heads exactly twice by dividing the number of favorable outcomes, 3, by the number of total possible outcomes, 8. 3/8