The remaining results from the survey are organized in the following table.
Consider the presented data to find the probabilities of the following scenarios.
The probability that the second book is a Geometry book given that the first book chosen is a History book equals
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
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
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
Considering these details, it can be concluded that of the bags containing forbidden items could trigger the alarm and 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|
|Forbidden and No Alarm|
|Not Forbidden and Alarm|
|Not Forbidden and No Alarm|
Finally, all the information can be shown on the tree diagram.
|Probabilities of the Events|
Take note that the sum of the probabilities is equal to which is the percentage of the bags that contain forbidden items.
There is about a chance that Mark's bag contains a forbidden item.
This probability is not close enough to 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 chance that Izabella's bag contains a forbidden item.
Since this probability is very small — less than — 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.
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
The possible outcomes are given by P(A) and the favorable outcomes by Therefore, the conditional probability formula can be obtained using the probability formula.
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.
To find the corresponding probabilities, take a look at the table.