If a random light bulb turns out to be defective, what is the probability that it was manufactured in Factory A?Īnswer: There are two random variables: Factory (A or B) and Defect (Yes or No). One percent of the light bulbs from Factory A are defective, while two percent of those from Factory B are defective. Question: You work for a lighting company which manufactures 60% of its light bulbs in Factory A and 40% in Factory B. The sensitivity and specificity are interpreted as follows:Īnd P(negative|¬cancer) = 0.97 ⃝c Alan Blair, 2017-20 The probability is called a prior, because it represents our estimate of the probability before we have done the test (or made some other observation). If a patient tests positive, what is the probability that they have the cancer?Īnswer: There are two random variables: Cancer (true or false) and Test (positive or negative). The test has a sensitivity of 98% and a specificity of 97%. Question: Suppose we have a test for a type of cancer which occurs in 1% of patients. P(Cause|Effect) = P(Effect|Cause)P(Cause) If P(B) ̸= 0, then the conditional probability of A given B is P(A|B)= P(A∧B) This is often useful for assessing the probability of an underlying cause

The formula for conditional probability can be manipulated to find a relationship when the two variables are swapped: Logically related events must have related probabilities For example, P(A∨B) = P(A)+P(B)−P(A∧B) P(X = 20.5) = 0.125 really meansĬOMP9444 20T2 Probability and Backprop Variations P induces a probability distribution for any random variable X:Į.g., P(Odd=true) = P(1)+P(3)+P(5) = 1 + 1 + 1 = 1 6 6 6 2 P(X = x) = U (x) = uniform density between 18 and 26 0.125 Probability for Continuous Variables (3.3)Ī random variable is a function from sample points to some range (e.g. e.g.P(dieroll<5)=P(1)+P(2)+P(3)+P(4)=1+1+1+1 =2ĬOMP9444 20T2 Probability and Backprop Variations 1ĬOMP9444 20T2 Probability and Backprop Variations 4ĬOMP9444 20T2 Probability and Backprop Variations 5 6 possible rolls of a die) Each ω ∈ Ω is a sample point / possible world / atomic eventĪ probability space or probability model is a sample space with an assignment P(ω) for every ω ∈ Ω such that 􏰈 Entropy and KL-Divergence (3.13) 􏰈 Cross Entropy (3.13)īegin with a set Ω – the sample space (e.g. 􏰈 Probability and Random Variables (3.1-3.2) 􏰈 Probability for Continuous Variables (3.3) 􏰈 Gaussian Distribution (3.9.3)

Dieroll coding