Conditional and Compound Probabilities


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The only information that the economist sees are the entries in the following table:. Previous Section.

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Table of Contents. Next Section. To learn the concept of independence of events, and how to apply it. Conditional Probability Suppose a fair die has been rolled and you are asked to give the probability that it was a five. Definition The conditional probability The probability of the event A taking into account the fact that event B is known to have occurred. Example 20 A fair die is rolled. Find the probability that the number rolled is a five, given that it is odd. Find the probability that the number rolled is odd, given that it is a five.

Example 21 In a sample of individuals under 40 who were or had previously been married, each person was classified according to gender and age at first marriage. E W H Total M 43 Example 22 Suppose that in an adult population the proportion of people who are both overweight and suffer hypertension is 0. Find the probability that the person selected suffers hypertension given that he is overweight. Find the probability that the selected person suffers hypertension given that he is not overweight. Compare the two probabilities just found to give an answer to the question as to whether overweight people tend to suffer from hypertension.

Definition Events A and B are independent Events whose probability of occurring together is the product of their individual probabilities. Example 23 A single fair die is rolled. Example 24 The two-way classification of married or previously married adults under 40 according to gender and age at first marriage in Note 3. Example 25 Many diagnostic tests for detecting diseases do not test for the disease directly but for a chemical or biological product of the disease, hence are not perfectly reliable. What is the probability that both test results will be positive? What is the probability that at least one of the two test results will be positive?


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Example 26 The specificity of a diagnostic test for a disease is the probability that the test will be negative when administered to a person who does not have the disease. A person who does not have the disease is tested for it using this procedure. What is the probability that the test result will be positive? A person who does not have the disease is tested for it by two independent laboratories using this procedure.

Example 27 The reliability of a system can be enhanced by redundancy, which means building two or more independent devices to do the same job, such as two independent braking systems in an automobile. Solution: Let D 1 denote the event that the contraband is detected by the first dog, D 2 the event that it is detected by the second dog, and D 3 the event that it is detected by the third. Probabilities on Tree Diagrams Some probability problems are made much simpler when approached using a tree diagram. Example 28 A jar contains 10 marbles, 7 black and 3 white. What is the probability that both marbles are black?

What is the probability that exactly one marble is black? What is the probability that at least one marble is black? Solution: A tree diagram for the situation of drawing one marble after the other without replacement is shown in Figure 3. Figure 3. Probabilities on Tree Diagrams The probability of the event corresponding to any node on a tree is the product of the numbers on the unique path of branches that leads to that node from the start.

If an event corresponds to several final nodes, then its probability is obtained by adding the numbers next to those nodes.

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Calculating conditional probability

Key Takeaways A conditional probability is the probability that an event has occurred, taking into account additional information about the result of the experiment. A conditional probability can always be computed using the formula in the definition. Sometimes it can be computed by discarding part of the sample space.

Find P B A. Determine whether or not A and B are independent. The probability that the roll is even. The probability that the roll is even, given that it is not a two. The probability that the roll is even, given that it is not a one. The probability that the second toss is heads. The probability that the second toss is heads, given that the first toss is heads. The probability that the second toss is heads, given that at least one of the two tosses is heads.

The probability that the card drawn is red. The probability that the card is red, given that it is not green. The probability that the card is red, given that it is neither red nor yellow.

Conditional Probability

The probability that the card is red, given that it is not a four. The probability that the card drawn is a two or a four. The probability that the card is a two or a four, given that it is not a one. The probability that the card is a two or a four, given that it is either a two or a three.

Dependent Events

The probability that the card is a two or a four, given that it is red or green. The probability that the family has at least two boys. The probability that the family has at least two boys, given that not all of the children are girls. The probability that at least one child is a boy. The probability that at least one child is a boy, given that the first born is a girl.

The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation A , B , C , or None and opinion on a bond issue: Affiliation Opinion Favors Opposes Undecided A 0. The following two-way contingency table gives the breakdown of the population of patrons at a grocery store according to the number of items purchased and whether or not the patron made an impulse purchase at the checkout counter: Number of Items Impulse Purchase Made Not Made Few 0.

An event that does not affect the occurrence of another subsequent event in a random experiment is an independent event. Both of the above examples are simple events events with single outcomes. When the occurrence of one event affects the occurrence of another subsequent event, the two events are dependent events. The concept of dependent events gives rise to the concept of conditional probability. It is the probability of an event given that another event has already occurred.

Independent and Dependent Events

The event that occurred earlier may affect the probability of the present event under consideration. Also, the event happening later can occur only after the given former event has taken place. Out of 10 customers in a box 4 are purchased red bulbs, 3 purchased green bulbs and 2 purchased both green and red. So, from the given formula, the probability that a consumer buying red bulb is also buying the green bulb is.

There are so many errors in two of the lectures that I have watched. The flow of the lectures are also inappropriate. Firstly you never defined what an event is. To generate a random value, using the weighted probability in the helper table, F5 contains this formula, copied down: Generally there is a very efficient algorithm called Belief Propagation, which gives exact results when the structure of the Bayesian Network is a singly connected tree there is only a single path between any two vertices in the undirected version of the graph.

When finding a conditional probability, you are finding the probability that an event A will occur, given that another event, event B, has occurred. Includes some examples of statements involving probability. Summary: The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The probability density function PDF of a random variable, X, allows you to calculate the probability of an event, as follows: For continuous distributions, the probability that X has values in an interval a, b is precisely the area under its PDF in the interval a, b.

My conditional probability of throwing a three if I know that the die that I threw is odd. Anyone writing a probability text today owes a great debt to William Feller, who taught us all how to make probability come alive as a subject matter. Chapter 2: Conditional Probability and Bayes formula We ask the following question: suppose we know that a certain event B has occurred.

Again, if the contingency table is available it is usually most efficient to take the appropriate values from the table, as shown above, as opposed to using these formulas. If E and F are events, then the probability of E given F is defined as.

Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities
Conditional and Compound Probabilities Conditional and Compound Probabilities

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