Suppose each day that you drive to work a traffic light that you encounter is either green with probability 3/8, red with probability 1/2, or yellow with probability 1/8, independent of the status of the light on any other day. If over the course of five days, G, Y, and R denote the number of times the light is found to be green, yellow, or red, respectively, what is the probability that P[G=2, Y=1, R=2] ? Also, what is the probability P[G=R]? [15%]. Hint: consider sequences possible in a five-day observations

a

b

Step-by-step explanation:

From the question we are told that

   The probability of the traffic light being green is

   The probability of the traffic light being red is  

   The probability of the traffic light being yellow is  

   The number of days of observation is n = 5

Generally the probability distribution function for a multinomial distribution is mathematically represented as

Generally the probability that P[G=2, Y=1, R=2]

=>

Generally the probability that P[G=R] is mathematically represented as

Here

Also  

a4 - 3a2 - 1

step-by-step explanation:

step   1   :

equation at the end of step   1   :

  ((a4) -   3a2) -   1

step   2   :

trying to factor by splitting the middle term

2.1     factoring   a4-3a2-1  

the first term is,   a4   its coefficient is   1 .

the middle term is,   -3a2   its coefficient is   -3 .

the last term, "the constant", is   -1