Final answer:
The probability of having the traffic light sequence with 2 greens, 1 yellow, and 2 reds over the course of five days is calculated using the binomial distribution and simplifies to 900/2048.
Step-by-step explanation:
The student has asked a question regarding the computation of a probability involving independent events over the course of multiple days. We are given that the traffic light can be green with probability 3/8, red with probability 1/2, and yellow with probability 1/8. The probability of encountering the light in a certain state on one day is independent of its state on any other day. To find P[G=2, Y=1, R=2], we will calculate the probability of two greens, one yellow, and two reds over five days.
Since the order of the lights encountered matters, we use the binomial distribution to calculate the probabilities of each scenario and then multiply them together. The probability can be computed as:
P(G = 2) x P(Y = 1) x P(R = 2)
This is calculated by taking the combinations for 2 greens, 1 yellow, and 2 reds in 5 days and multiplying them by the respective probabilities:
(5 choose 2) x (3/8)^2 x (1/8)^1 x (5 choose 1) x (1/2)^2
Plugging in the values we get:
10 x (9/64) x (1/8) x 10 x (1/4)
This simplifies to:
10 x 9 x 1 x 10 / 64 x 8 x 4
And further to:
900 / 2048
Hence, the probability P[G=2, Y=1, R=2] is 900/2048.