Consider the following two events.
|Event||The DNA from a crime scene is concluded to be the DNA of the defendant.|
|Event||The defendant is not guilty.|
If the defendant is not guilty — event occurs — then the probability that their DNA was found on the crime scene is very small. Therefore, the conditional probability is small. Often, based on this information, the prosecutor claims that the probability that the defendant is not guilty given that occurs is also small.This reasoning is based on the assumption that is equal to Is this assumption, and therefore the reasoning for it, correct or incorrect? Why?
The probability of being infected with a certain disease is There is a test used to detect this disease. If someone is infected, the test comes back positive times out of cases. If someone is not infected, the test comes back positive time out of cases. Express the answers as decimal numbers approximated to one significant figure.
Consider two dependent events and The conditional probability of given is the ratio of the probability of the intersection of and to the probability of The Multiplication Rule of Probability is obtained by multiplying both sides of the above formula by and using the Symmetric Property of Equality. Following similar reasoning, an equivalent form of the rule can be obtained.
A tree diagram will be made to answer the questions. The root node of the tree represents the event of choosing a number from The chosen number can be either from or from
The probability of each outcome should be written on the corresponding branch. Set has elements, has elements, and the sets do not have any elements in common. With this information the probability that is an element of and that is an element of can be calculated. Now the probabilities will be added to the diagram.
Set has elements. of them are positive and of them are negative. Therefore, knowing that is an element of the probability that it is positive is and the probability it is negative is also
Likewise, has elements: of them positive and of them negative. Knowing that is from the probability it is positive is and the probability it is negative is
The probability that is an element of and positive can be calculated by multiplying the probabilities along the corresponding branch of the tree diagram.
Similarly, the probability that is an element of and negative can be found.
Dylan participates in a school lottery that consists of two stages. He starts by drawing a ticket from a hat. This ticket tells Dylan from which of three boxes he will be drawing a second ticket. There are tickets in the hat: for box A, for box B, and for box C.
First make a tree diagram. The root node of the tree represents the event of drawing a ticket from the hat. There are tickets in the hat. From those, correspond to box A, correspond to box B, and correspond to box C. With this information, the probabilities for the first three branches of the tree can be written.
A ticket drawn from any of the boxes can be a winning or a loosing ticket. Each box contains tickets. Box A has winning ticket, box B has and box C With this information, the probabilities of drawing a winning ticket from each box can be written.
The sum of the probabilities of the branches that come out of the same node is equal to Knowing this, the probabilities of not drawing a winning ticket from each box can be calculated.
There are three possible outcomes in which Dylan draws a winning ticket.
How can the probability that the marble is from box A and it is orange be calculated using the given information?
marble from box Aand
orange marble,it can be rewritten using the Multiplication Rule of Probability. The above expression can be substituted into the formula for the probability that the marble is from box A knowing it is orange. The conditional probability has been expressed in terms of the known probabilities. Now, the values of the known probabilities can be substituted into the formula.
First, the given information will be represented in a tree diagram.
The sum of the probabilities of branches coming out of the same node is always Using this fact, the tree diagram can be completed.
The tree diagram gathers all of the known information. Now, the desired unknown probabilities will be calculated one by one.
By the definition of conditional probability, the conditional probability that a person is infected given their test result is negative can be written as a ratio. The probability of the intersection of two events can be rewritten using the Multiplication Rule of Probability. Both probabilities in the numerator are represented in the tree diagram.
The probability that the test comes back negative can be found using the tree diagram as well. To do so, the probabilities of all outcomes that include receiving a negative test result should be added. There are two such outcomes. These are
infected with a negative test result and
not infected with a negative test result.
The probability that a person is not infected given the test result is positive can be found by reversing the conditional probability. It is known that The remaining two probabilities in the above expression are represented on the tree diagram.
The two given conditional probabilities have been reversed. Based on that, the example values of and can be compared.
It is seen that and are not equal.