Example:Meeting The event that Romeo and Juliet will meet is the shaded region. ■ Its probability is calculated to be 7/16. =1-the area of the two 1/ unshaded triangles =1-2·(@)·(/2 1/4 =7/16. M={x,y)x-y川≤40≤x≤1,0≤y≤1
Example: Meeting The event that Romeo and Juliet will meet is the shaded region. Its probability is calculated to be 7/16. = 1 − the area of the two unshaded triangles = 1 − 2 ⋅ 3 4 · 3 4 /2 = 7/16. 𝑀 = 𝑥, 𝑦 : 𝑥 − 𝑦 ≤ 1 4 , 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1
Properties of Probability Laws Consider a probability law,and let A,B, and C be events. 1.IfA∈B,then P(A)≤P(B). 2.P(AUB)=P(A)+P(B)-P(AOB). 3.P(AUB)≤P(A)+P(B). 4.P(AUBUC)=P(A)+P(AOB)+ P(AcBC). A is the complement of A
Properties of Probability Laws Consider a probability law, and let 𝐴, 𝐵, and 𝐶 be events. 1. If 𝐴 ⊆ 𝐵, then 𝑃 𝐴 ≤ 𝑃 𝐵 . 2. 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵 . 3. 𝑃(𝐴 ∪ 𝐵) ≤ 𝑃 𝐴 + 𝑃 𝐵 . 4. 𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) = 𝑃(𝐴) + 𝑃(𝐴 𝑐 ∩ 𝐵) + 𝑃(𝐴 𝑐 ∩ 𝐵 𝑐 ∩ 𝐶). 𝐴 𝑐 is the complement of 𝐴
Content Sets. Probabilistic models. Conditional probability. Total Probability Theorem and Bayes'Rule. Independence. Counting
Content Sets. Probabilistic models. Conditional probability. Total Probability Theorem and Bayes’ Rule. Independence. Counting
Partial information ■( Conditional probability provides us with a way to reason about the outcome of an experiment,based on partial information. Example:In an experiment involving two successive rolls of a die,you are told that the sum of the two rolls is 9.How likely is it that the first roll was a 6? Example:How likely is it that a person has a certain disease given that a medical test was negative? Example:A spot shows up on a radar screen.How likely is it to correspond to an aircraft?
Partial information Conditional probability provides us with a way to reason about the outcome of an experiment, based on partial information. Example: In an experiment involving two successive rolls of a die, you are told that the sum of the two rolls is 9. How likely is it that the first roll was a 6? Example: How likely is it that a person has a certain disease given that a medical test was negative? Example: A spot shows up on a radar screen. How likely is it to correspond to an aircraft?
Conditional Probability In previous examples,we know that the outcome is within some given event B. We wish to quantify the likelihood that the outcome also belongs to some other event A. We seek to construct a new probability law that takes into account the available knowledge: a probability law that specifies the conditional probability of A given B
Conditional Probability In previous examples, we know that the outcome is within some given event 𝐵. We wish to quantify the likelihood that the outcome also belongs to some other event 𝐴. We seek to construct a new probability law that takes into account the available knowledge: a probability law that specifies the conditional probability of 𝐴 given 𝐵