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- Probability, Conditional Probability, and Bayes’ Rule

*You can download the Bayes Theorem conditional probability and its applications-examples PDF or you can go through the details below. Conditional probability is used in case of events which are not independent.*

It is difficult to find an explanation of its relevance that is both mathematically comprehensive and easily accessible to all readers. It builds on Meehl and Rosen's classic paper, by laying out algebraic proofs that they simply allude to, and by providing extremely simple and intuitively accessible examples of the concepts that they assumed their reader understood.

Although it is simple in its conception, Baye's Rule can be fiendishly difficult for beginners to understand and apply. A great deal has been written about the importance of conditional probability in diagnostic situations. However, there are, so far as I know, no papers that are both comprehensive and simple.

Most writing on the topic, particularly in probability textbooks, assumes too much knowledge of probability for diagnosticians, losing the clinical reader by alluding to simple proofs without giving them. Many introductory psychometrics textbooks err on the other side, either ignoring conditional probability altogether, or by considering it in such a cursory manner that the reader has little chance to understand what it is and why it matters.

This paper is intended to fill the void between simplicity and thoroughness. Many readers find this paper very difficult to understand, in part because the authors do make mathematical claims without providing any detailed explanation of where they came from. The present paper frames Meehl and Rosen's claims with a much more basic introduction than they give, and fills in some simple proofs to which they only allude. The first section consists of a general introduction to understanding conditional probabilities.

Conditional probabilities are those probabilities whose value depends on the value of another probability. Such probabilities are ubiquitous. For example, we may wish to calculate the probability that a particular patient has a disease, given the presence of a particular set of symptoms. The probability of disease may be more or less close to certain, depending on the nature and number of symptoms.

We will certainly wish to take into account a patient's relevant prior history with medication e. Or we may wish to take into account factors such as defensiveness that might impact on a success in psychotherapy before we begin that therapy Zanarini et al. More generally, restating all these specific cases in a more abstract way, we may wish to calculate the probability that a given hypothesis is true, given a diverse set of evidence say, results from several diagnostic instruments for or against it.

Hypothesis testing is just one way of assigning weight to belief. A very simple example of conditional probability will elucidate its nature. Consider the question: How likely is that you would win the jackpot in a lottery if you didn't have a lottery ticket? It should be obvious that the answer is zero — you certainly could not win if you didn't even have a ticket. It may be equally obvious that you are more likely to win the lottery the more tickets you buy.

So the probability of winning a lottery is really a conditional probability, where your odds of winning are conditional on the number of tickets you have purchased. If you have zero tickets, then you have no chance of winning. With one ticket, you have a small chance to win. With two tickets, your odds will be twice as good.

One thing that sometimes confuses students of probability is the fact that all probability problems are really conditional.

This observation sheds light on what conditionality actually does. An appropriate way of thinking about conditional probability is to understand that a conditional limits the number and kind of cases you are supposed to consider. To see that this is so, consider the following simple question:.

Three tall and two short men went on a picnic with four tall and four short woman. What is P Tall Female , the probability that a person is tall, given that the person is female? The solution to this problem may be immediately obvious, but it is worth working through a few ways of solving it.

These are all formally the same, though they may appear to be different. The first way is just to turn the question into a very simple non-conditional question that we know how to solve. What is the probability that a woman who went on the picnic was tall? Here comes the tricky part. This diagram makes clear what the question is asking: What is the ratio of people who are both tall and female top left cell to people who are female sum of left column?

We can re-state this and solve the problem in a third way by asking: What is the ratio of the probability that a person is both female and tall to the probability that a person is female?

To see why, consider the concrete example again. There were 13 people on the picnic. Before we look at how the math works, let's introduce the rule itself. Bayes was a minister interested in probability and stated a form of his famous rule in the context of solving a somewhat complex problem involving billiard balls that need not concern us here.

This paper concerns itself almost entirely with the simplest form, which covers the cases in which two sets of mutually exclusive possibilities A and B are considered, and where the total probability in each set is 1.

The simplest case covers many diagnostic situations, in which the patient either has or does not have a diagnosable condition possibility set A and either has or does not have a set of symptoms possibility set B.

P A is called the marginal or prior probability of A , since it is the probability of A prior to having any information about B. Similarly, the term P B is the marginal or prior probability of B. Because it does depend on having information about B, the term P A B is called the posterior probability of A given B. In the third solution to the example above, we solve for the probability of being female, given that you are tall, by considered the ratio of those who were tall and female to those who were female:.

From this it should be evident, by equating the numerators of the two equations above, that:. Let's see how the definition agrees with this answer.

Let's see why using the same example. The first calculation picks out the cell of tall females by column. The second picks it out by row. It doesn't matter if you concern yourself with females who are tall or tall people who are females — in the end you must get to the same answer if you want to know about people who are both tall and female.

A tall female person is also a female tall person. If we already know P A B , then we don't need to compute it. If we don't know it, then it will not help us to include it in the equation we will use to calculate it. In diagnostic cases where were are trying to calculate P Condition Symptom we often know P Symptom Condition , the probability that you have the symptom given the condition, because this data has been collected from previous confirmed cases.

However, its implications are often unexpected. Many studies have shown that people of all kinds — even those who are trained in probability theory — tend to be very poor at estimating conditional probabilities.

It seems to be kind of innate incompetence in our species. Let us consider a concrete example given in Meehl and Rosen , from which much of the discussion in this section is drawn. What is the chance that a randomly selected person with a positive result actually has the disease? How can this be? Since so few people actually have the disease, the probability of a true positive test result is very small. It is swamped by the probability of a false positive result, which is fifty times larger than the probability of a true positive result.

You can concretely understand how the false positive rate swamps the true positive rate by considering a population of 10, people who are given the test.

Many cases are subtle. Consider another case cited by Meehl and Rosen This involved a test to detect psychological adjustment in soldiers. The authors of the instrument validated their test by giving it to soldiers known to be well-adjusted, and 89 soldiers known to be mal-adjusted. However, they failed to take into account base rates. The test is still better than guessing that everyone is maladjusted. Of course clinicians prefer to make diagnoses that are more likely to be right than wrong.

We can state this desire more formally by saying that we prefer the fraction of the population that is diagnosed correctly to be greater than the fraction of the population that is diagnosed incorrectly.

Mathematically this leads to a useful conclusion in the following manner:. We need the ratio of positive to negative base rates to be greater than the ratio of the false positive rate to the true positive rate, if we want to be more likely to be right than wrong.

This can be a handy heuristic because it allows us to calculate the minimum proportion of the population we are working with that needs to be diseased in order for our diagnostic methods to be useful. In the example above, the ratio of false positive to true positive rates is 0.

This means that the test can only be useful — in the sense of having a positive diagnosis that is more likely to be true than false — when it is used in settings in which the ratio of the maladjusted people positive base rate to the number of people who are not maladjusted negative base rate is at least 0.

Again we can consider one example from Meehl and Rosen The calculation above says that this test will only be reliable if the ratio of brain-damaged to non-brain-damaged people is greater than 0. If we are using the test in a setting which has a lower ratio of brain damaged people, we will run in to the problem described above, in which we find that the base rates have made it more likely that we are wrong than right when we make a diagnosis.

They therefore suggest 8 as the optimal cut-off. With that cut-off, the test has a true positive rate sensitivity of 0. It has a true negative rate specificity of 0. The ratio of false to true positives is thus 0. With the prescribed cut-off point, the test will only predict violence correctly if at least With a higher cut-off of 16, the true positive rate is just 0. This gives a ratio of false to true positives of 0. In this case, the mathematical result is somewhat equivocal because of the unequal costs of making false positive and false negative identifications.

The rate of identifying future violence is certainly very poor with the prescribed cut-off of 8. The ratio of false to true positives shows that if a person uses this cut-off, he will do only a little better than he would if he predicted who will be violent by flipping a coin, since using the cut-off will make him wrong Sometimes we have pragmatic reasons to prefer one kind of inaccuracy to another.

Note that Meehl's heuristic does not mean that the true population base rate must be as high as the calculation prescribes — it is sufficient for the base rate of the subpopulation to which the test is exposed to be high enough.

For example, Fontaine et al. This ability to skew true diagnosis rates in a favorable direction by pre-selecting subjects has important implications.

It is difficult to find an explanation of its relevance that is both mathematically comprehensive and easily accessible to all readers. It builds on Meehl and Rosen's classic paper, by laying out algebraic proofs that they simply allude to, and by providing extremely simple and intuitively accessible examples of the concepts that they assumed their reader understood. Although it is simple in its conception, Baye's Rule can be fiendishly difficult for beginners to understand and apply. A great deal has been written about the importance of conditional probability in diagnostic situations. However, there are, so far as I know, no papers that are both comprehensive and simple. Most writing on the topic, particularly in probability textbooks, assumes too much knowledge of probability for diagnosticians, losing the clinical reader by alluding to simple proofs without giving them.

S = 1 2 3. 4 5 6. Let A = 6 appears. B = an even number appears. So. P(A) = 1. 6. P(B) = 1. 2. Lecture 4: Conditional Probability and Bayes' Theorem.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. I know the Bayes rule is derived from the conditional probability.

Most users should sign in with their email address. If you originally registered with a username please use that to sign in. To purchase short term access, please sign in to your Oxford Academic account above. Don't already have an Oxford Academic account?

Sign in. Conditional probability is the sine qua non of data science and statistics. The important point in data science is not the equation itself, the application of this equation to the verbal problem is more important than remembering the equation.

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Statistics for Bioengineering Sciences pp Cite as. If statistics can be defined as the science that studies uncertainty, then probability is the branch of mathematics that quantifies it. However, the formal, precise definition of probability is elusive. There are several competing definitions for the probability of an event, but the most practical one uses its relative frequency in a potentially infinite series of experiments.

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1. Bayes' Theorem by Mario F. Triola. The concept of conditional probability is introduced in Elementary Statistics. We noted that the conditional probability of an.

This is useful in practice given that partial information about the outcome of an experiment is often known, as the next example demonstrates. Continuing in the context of Example 1. So, knowing that at least one tails was recorded, i.

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## Celalophugh

Be able to use the multiplication rule to compute the total probability of an event. 4. Be able to check if two events are independent. 5. Be able to use Bayes'.