10 Recall that Batman failed his superhero test despite havi

10. Recall that Batman failed his superhero test, despite having a good feeling about it, the Joker decided to reprogram all of the traffic lights in Gotham. They now randomly go from one color to the next after a fixed time of 30 seconds. He programmed them to change independently according to the following transition matrix T. 1 0 .9 0 4 .6 where the colors are ordered Green, Yellow, Red (i.e p2,3 5 represents the probability the light changes from yellow to red) (a) If the light is yellow now, what is the probability that it will be green next? (b) If the light is yellow now, what is the probability that it will be red 2 after 2 changes? (i. find p (2) (2) and

Solution

(a)

P(light turns green | it is yellow) is p₂₁ in the given transition matrix.

p₂₁=0.2

(b)

Let X_i denote the state of light at ith instant time of time starting from present. At t₀ , X₀=2. Also, note that X_i+1 depends only on X_i.

P(X₂=3 | X₂=2)=P(X₂=3,X₁=1| X₀=2)+P(X₂=3,X₁=2| X₀=2) +P(X₂=3,X₁=3| X₀=2)

=P(X₂=3 | X₁=1, X₀=2)*P(X₁=1| X₀=2)+P(X₂=3 |X₁=2, X₀=2)*P(X₁=2| X₀=2)+P(X₂=3 | X₁=3, X₀=2)*P(X₁=3| X₀=2)

=P(X₂=3 | X₁=1)*P(X₁=1| X₀=2)+P(X₂=3 |X₁=2)*P(X₁=2| X₀=2)+P(X₂=3 | X₁=3)*P(X₁=3| X₀=2)

=0.9*0.2 + 0.5*0.3 + 0.6*0.5

=0.18+0.15+0.30

=0.63