Answer:
a) 11.6%
b) 0.52%
c) 1.75
Step-by-step explanation:
For each light, there are only two possible outcomes. Either it is red, or it is not. The probability of a light being red is independent from other lights. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
There are 5 traffic signals between your home and work.
This means that ![n = 5](https://tex.z-dn.net/?f=n%20%3D%205)
Each is red with probability 0.35
This means that ![p = 0.35](https://tex.z-dn.net/?f=p%20%3D%200.35)
a) the probability of encountering no red lights
This is P(X = 0). So
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 0) = C_{5,0}.(0.35)^{0}.(0.65)^{5} = 0.116](https://tex.z-dn.net/?f=P%28X%20%3D%200%29%20%3D%20C_%7B5%2C0%7D.%280.35%29%5E%7B0%7D.%280.65%29%5E%7B5%7D%20%3D%200.116)
So the answer for a is 11.6%
b) the probability of encountaring only red lights
This is P(X = 5)
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 5) = C_{5,5}.(0.35)^{5}.(0.65)^{0} = 0.0052](https://tex.z-dn.net/?f=P%28X%20%3D%205%29%20%3D%20C_%7B5%2C5%7D.%280.35%29%5E%7B5%7D.%280.65%29%5E%7B0%7D%20%3D%200.0052)
So the answer for b is 0.52%.
c) the expected number of red lights you will encounter?
The expeced number of the binomial distribution is given by:
![E(X) = np](https://tex.z-dn.net/?f=E%28X%29%20%3D%20np)
So
![E(X) = 5*0.35 = 1.75](https://tex.z-dn.net/?f=E%28X%29%20%3D%205%2A0.35%20%3D%201.75)