Conditional Probability, Total Probability Theorem and ... - Berlin Chen

UBiiSome Examples Using Total Probability Theorem (3/3)• Example 1.15. Alice is taking a probability class and at the end ofeach week she can be either up-to-date or she may have fallenbehind. If she is up-to-date in a given week, the probability that shewill be up-to-date (or behind) in the next week is 0.8 (or 0.2,respectively). If she is behind in a given week, the probability thatshe will be up-to-date (or behind) in the next week is 0.4 (or 0.6,respectively). Alice is (by default) up-to-date when she starts theclass. What is the probability that she is up-to-date after three weeks?: up - to - date: behindPPPU3 PU2 PU3 U 2 PB2PU3 B2PU2 0.8 PB2U2 PU1PU2 U1PB1PU2 B1PU10.8 PB1BPBPBU PBPBB PU0.2 PB0.2 PPPPU10.8,PB10.2 PU0 U2 0.80.8 0.2 0.4 0.72P Ui1B20.80.2 0.2 0.6 0.28P Bi1U0.72 0.8 0.28 0.4 0.68831As we know that2112111.01Recursion formuleaP0.40.46 PUi 0.8 PBi PUi 0.2 PBiU0.8, PB0. 211 0.4 0.6Probability-Berlin Chen 18