Question

Question No 10: Suppose at Islamabad Kashmir Highway, the traffic engineers have coordinated the timings of two traffic signals to encourage a run of green signals. In particular, the timing was designed so that with probability 0.8 a driver will find second signals’ light to have the same color as the first. Assuming the first signal is equally likely to be red or green,
i. What is the probability that the second signal is green?
ii. What is the probability that the driver at least waits at one of the signal?
iii. What is the probability of a first green signal, given that the second signal is red?

NOTE: kindly solve the question and also explain which approach u used and which keyword led you to decide which approach to go for . kindly explain all the steps as i have trouble understanding probability.

Answer 1

let

G1,G2 represent first and 2nd green signal respectively

R1,R2 represent the first and 2nd red signal respectively

so given that

P(G1)=P(R1) =0.5

P(G2|G1)=0.8 P(G2|R1)=0.2 P(R2|G1)=0.2 P(R2|R1) =0.8 (these are probabilities that 2nd signal is red or green given that first signal is red or given)

a)

now we have to find P(G2)

now

P(G2) =P(G2|G1)*P(G1) +P(G2|R1)*P(R1)

=0.8*0.5+0.2*0.5

=0.5

b)

P( driver waits at least at one signal) =P(getting at least one red signal)

=P(first red and 2nd green signal) +P(first green signal and 2nd red signal) +P(both signals red)

=P(G2|R1)*P(R1)+P(R2|G1)*P(G1)+P(R2|R1)*P(R1)

=0.2*0.5+0.2*0.5+0.8*0.5

=0.6

c)

we have to find P(G1|R2)=?

now using bayes rule