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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 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.
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}.
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.
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.
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.
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.
Event | Number of Outcomes |
---|---|
Flipping a coin | 2 |
Rolling a die | 6 |
Flipping a coin and rolling a die | 2Ă—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.
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 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.
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.
Rewrite 18 as 2â‹…9
Cancel out common factors
Simplify quotient
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.
Rewrite 20 as 5â‹…4
Cancel out common factors
Simplify quotient
We want to find the probability of getting at least one tails when flipping a coin three times in a row. We first need to know the sample space of the experiment before finding the probability. We can start by drawing a tree diagram to help us find the sample space. Begin by 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.
Let's do the same for the third throw.
We can now write the sample space of the experiment.
The sample space consists of 8 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.
Note that we have 7 different ways of getting at least one tails. Find the probability of this event by dividing 7 by the number of possible outcomes. 7/8
We are told that a six-sided die is rolled two times, and we are asked 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 plenty of outcomes, making a tree diagram would take a lot of space. Instead, we multiply the number of possible outcomes of each roll. 6 * 6 = 36 The given experiment has 36 different outcomes. Since we are interested in the sum of the rolls, we will inspect in which cases the sum of rolled numbers is 11. This can only happen when we roll a combination of the numbers 5 and 6. 5+6 &= 11 6+5 &= 11 Note that if either die rolls less than 5, then it is not possible to obtain a sum of 11. This means that there are only 2 favorable outcomes. Knowing this, we can find the requested probability by dividing the number of favorable outcomes by the number of all possible outcomes. 2/36 Notice that the above fraction can be simplified.
The probability of getting the die to add up to exactly 11 in two throws is 118.
We are told that Magdalena's cat is about to give birth to three kitties. To find the probability that she gives birth to at least one male and a female kitty, we will first write the sample space of this experiment. Let's make a tree diagram! We begin by listing the possible outcomes of the first born kitty.
We now list the possibilities of the second kitty and use segments to connect to their sibling.
The third birth is listed in a similar way.
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 kitty. We will highlight these outcomes.
There are 6 out of 8 possible outcomes where there is at least one male and one female kitty. The quotient of these numbers is the probability of this compound event. 6/8 Notice that we can simplify this fraction.
We are told that the password is made of four digits. Recall that the 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, then another 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 four times by itself. 10 * 10 * 10 * 10 = 10 000 There are 10 000 possible four-digit passwords. Only 1 of them is the correct one. This information allows us to find the probability of getting the password in the first try. 1/10 000 The probability of getting the password right on the first try is 110 000. It is better if Magdalena calls her sister before the laptop gets blocked!