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| 8 Theory slides |
| 6 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.
The remaining results from the survey are organized in the following table.
Consider the presented data to find the probabilities of the following scenarios.
Substitute values
c/da/b=ba⋅cd
Multiply fractions
ba=b/5a/5
Substitute values
c/da/b=ba⋅cd
Multiply fractions
ba=b/15a/15
The probability that the second book is a Geometry book given that the first book chosen is a History book equals P(H)P(H∩G). |
The previous statement can also be rewritten in terms of H and G as follows.
The probability that event G happens given that event H happened equals P(H)P(H∩G). |
Similarly, the second probability found in part B equals the probability found in part D. This leads to write the following relation.
The probability that the first book is a History book, given that the second book is a Geometry book equals P(G)P(H∩G). |
As before, the previous statement can be rewritten in terms of H and G.
The probability that event H happens, given that event G happened equals P(G)P(H∩G). |
Considering these details, it can be concluded that 2% of the bags containing forbidden items could trigger the alarm and 96% of the bags that do not have forbidden items could not trigger the alarm.
Now, using the percentages in the branches, the number of bags for each event can be found.
Forbidden and Alarm | 350⋅98%=343 |
---|---|
Forbidden and No Alarm | 350⋅2%=7 |
Not Forbidden and Alarm | 4650⋅4%=186 |
Not Forbidden and No Alarm | 4650⋅96%=4464 |
Finally, all the information can be shown on the tree diagram.
Calculate quotient
Convert to percent
Probabilities of the Events | ||
---|---|---|
P(Alarm and Forbidden) | 5000343=6.86% | |
P(No Alarm and Forbidden) | 50007=0.14% |
Take note that the sum of the probabilities is equal to 7%, which is the percentage of the bags that contain forbidden items.
Alarm.
Calculate quotient
Round to 4 decimal place(s)
Convert to percent
Calculate quotient
Round to 4 decimal place(s)
Convert to percent
There is about a 64.84% chance that Mark's bag contains a forbidden item. |
This probability is not close enough to 100% to ensure that Mark's bag contains a forbidden item. Therefore, it is doubtful — but possible — that Mark's bag contains a forbidden item. Next, recall the answer found in Part D.
There is about a 0.16% chance that Izabella's bag contains a forbidden item. |
Since this probability is very small — less than 1% — it is almost certain that Izabella does not have forbidden items in her bag — but still possible.
Conditional probability is the measure of the likelihood of an event B occurring, given that event A has occurred previously. The probability of B given A is written as P(B∣A). It can be calculated by dividing the probability of the intersection of A and B by the probability of A.
P(B∣A)=P(A)P(A and B),where P(A)=0
The intuition behind the formula can be visualized by using Venn Diagrams. Consider a sample space S and the events A and B such that P(A)=0.
Assuming that event A has occurred, the sample space is reduced to A.
This means that the probability that event B can happen is reduced to the outcomes in the intersection of A and B, that is, to those outcomes in A∩B.
The possible outcomes are given by P(A) and the favorable outcomes by P(A∩B). Therefore, the conditional probability formula can be obtained using the probability formula.
P(B∣A)=P(A)P(A and B)
Diego's generous father has finished doing laundry and put Diego's T-shirts along with those of his big brother into the same ol’ basket. There are orange, blue, and red T-shirts in the basket, of which four are S-sized and eight are M-sized.
P(S and O)=121, P(O)=125
c/da/b=ba⋅cd
Multiply fractions
ba=b/12a/12
P(S and O)=121, P(S)=31
c/da/b=ba⋅cd
Multiply fractions
ba=b/3a/3
P(S and B)=61, P(B)=31
c/da/b=ba⋅cd
Multiply fractions
ba=b/3a/3
P(S and B)=61, P(S)=31
c/da/b=ba⋅cd
Multiply fractions
ba=b/3a/3
Find the required conditional probability and round it to two decimal places.
To find the corresponding probabilities, take a look at the table.
P(Penguin and France)=381, P(France)=190157
c/da/b=ba⋅cd
Multiply fractions
Calculate quotient
Convert to percent
Round to nearest integer
P(Penguin and France)=381, P(Penguins)=387
c/da/b=ba⋅cd
Multiply fractions
Calculate quotient
Convert to percent
Round to nearest integer
P(Antarctica and No Penguin)=1903, P(No Penguin)=3831
c/da/b=ba⋅cd
Multiply fractions
Calculate quotient
Convert to percent
Round to nearest integer
Suppose a voter poll is conducted in three states. The following information is recorded.
Let's start by illustrating the information about the number of people living in the three states. Of 100 % of the voters, 40 % live in State A, 25 % live in State B, and 35 % live in State C. Let's put this information in a table, in the total row.
To fill the cells in the first row, we multiply the percentages that support the liberal candidate in each state by the corresponding percent of people living in that state. P(A and Yes)&= 0.40(0.5)=0.20 P(B and Yes)&= 0.25(0.6)=0.15 P(C and Yes)&= 0.35(0.2)=0.07 Let's add this to the diagram and then sum all the values for liberal voters across the states. This gives the percentage of liberal voters in the three states.
As we can see, 42 % of all voters in the three states support the liberal candidate. Using the information from the table, we can determine the probability that a liberal voter lives in state B. To do so, we divide the percentage of liberal voters in State B by the percentage of liberal voters.
There is a 36 % probability that a liberal voter lives in state B.
There are three potential events for each part that is manufactured. Good& - 95 % Slightly defective& - 1 % Totally defective& - 4 % We know that the totally defective parts are thrown away. We want to determine the probability that a part is good given that it is shipped. The percentage of parts shipped equals the percentage of parts that were not totally defective. Since 4 % of all parts where totally defective, it must be that 96 % of all parts are shipped. If we label the event that a part gets shipped as S, we can write the following conditional probability. P(G|S)=P(G and S)/P(S) From the exercise, we know that 95 % of parts are good and we have also figured out that 96 % of all parts are shipped. Now we can calculate the conditional probability.
As we can see, about 99 % of all parts that are shipped are good.