Assume that there is a rare disease. It occurs in the population at a rate of 1 per 1000; that is, P[disease] = 0.001. There is a test for the disease that has an error rate of 5% for in positive and negative senses. That is, if a patient has the disease, there is 95% probability that the test will confirm that status and 5% probability that the test will be negative. Conversely, if the patient does not have the disease, there is 5% probability that the test will indicate erroneously that the disease is present and 95% probability that the test will confirm that the disease is not present. Let PT indicate a positive test, D indicate the disease is present, and a bar over a symbol indicates the negative, so this can all be summarized as Po(D) = 0.001 P(PT|D) = 0.95 P(PT|D) = 0.95 P(PT|D) = 0.05 P(PT|D) = 0.05
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![Assume that there is a rare disease. It occurs in the population at a rate of 1 per 1000; that is, P[disease] = 0.001. There is a test
for the disease that has an error rate of 5% for in positive and negative senses. That is, if a patient has the disease, there is 95%
probability that the test will confirm that status and 5% probability that the test will be negative. Conversely, if the patient does
not have the disease, there is 5% probability that the test will indicate erroneously that the disease is present and 95% probability
that the test will confirm that the disease is not present. Let PT indicate a positive test, D indicate the disease is present, and a bar
over a symbol indicates the negative, so this can all be summarized as
Po(D) = 0.001
P(PT|D) = 0.95
P(PT|D) = 0.95
P(PT|D) = 0.05
P(PT|D) = 0.05
The disease is serious, and the treatment is difficult and expensive, so a decision either to treat or not to treat the disease is not
trivial. A patient takes the test and receives a positive result. What is the probability that the patient has the disease?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862ea95c-ff68-4701-a33f-b6ae05be34dd%2F9bd12244-eb48-48ce-9f64-812cd852f7ef%2Flqsb1k9_processed.jpeg&w=3840&q=75)
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