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Here are a few recommended readings before getting started with this lesson.
Consider the following two events.
Event A | The DNA from a crime scene is concluded to be the DNA of the defendant. |
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Event B | The defendant is not guilty. |
Conversely, if A and B are dependent events, a rearrangement of the Conditional Probability Formula can be used to find the probability of the intersection of the events.
Outcomes that should be included in the tree diagram are a∈C, a∈D, a>0, and a<0.
A tree diagram will be made to answer the questions. The root node of the tree represents the event of choosing a number a from C∪D. The chosen number can be either from C or from D.
The probability of each outcome should be written on the corresponding branch. Set C has 6 elements, D has 4 elements, and the sets do not have any elements in common. With this information the probability that a is an element of C and that a is an element of D can be calculated.The other outcomes that need to be considered to answer the questions are that a can be positive or negative.
Set C has 6 elements. 3 of them are positive and 3 of them are negative. Therefore, knowing that a is an element of C, the probability that it is positive is 63 and the probability it is negative is also 63.
Likewise, D has 4 elements: 3 of them positive and 1 of them negative. Knowing that a is from D, the probability it is positive is 43 and the probability it is negative is 41.
The probability that a is an element of C and positive can be calculated by multiplying the probabilities along the corresponding branch of the tree diagram.
Similarly, the probability that a is an element of D and negative can be found.
For the following questions, approximate the answers to two decimal places.
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.
Substitute values
Multiply fractions
Cancel out common factors
Simplify quotient
b/12a/12=ba
Calculate quotient
For the following questions, approximate the answers to two decimal places.
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 1. 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.
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
.
Substitute values
Multiply
Use a calculator
\RoundSigDig{1}
P(negative)=0.99980301
Subtract term
Substitute values
Multiply
Use a calculator
\RoundSigDig{1}
Substitute values
Multiply
Use a calculator
\RoundSigDig{1}
The two given conditional probabilities have been reversed. Based on that, the example values of P(A∣B) and P(B∣A) can be compared.
P(positive|infected) | P(infected|positive) |
---|---|
0.97 | 0.5 |
P(positive|not infected) | P(not infected|positive) |
0.0001 | 0.5 |
It is seen that P(A∣B) and P(B∣A) are not equal.