3. A traffic light at a certain intersection is green 50% of the time, yellow 10

Question

3. A traffic light at a certain intersection is green 50% of the time, yellow 10% of the time, and red 40% of the time. A car approaches this intersection once each day. a. Let X represent the number of days that pass up to and including the first time a car encounters a red light. i. State the distribution of X and its parameter(s).(3 points) ii. Find P(X S 3). (4 points) ii. Find P(X 2 7). (4 points) b. Let Y denote the number of days up to and including the third day on which a red light is encountered. i. State the distribution of Y and its parameter(s). (3 points) ll. Find P(Y= 7). (4 points) c. What is the probability that in a sequence of 10 days, four green lights, one yellow light and five red lights are encountered? (4 points)

Explanation / Answer

P(G) = 0.5 , P(Y) = 0.1 and P(R) = 0.4

(a)
i) This is an example of geometric distribution with p = 0.4

ii)
P(X <= 3) = 1 - (1 - p)^3 = 1 - (1 - 0.4)^3 = 0.784

iii)
P(X >= 7) = 1 - P(X <= 6)
P(X <= 6) = 1 - (1 - p)^6 = 1 - (1 - 0.4)^6 = 1 - 0.9533 = 0.0467

(c)
Required probability, P = 10C4*(0.5)^4 * 6C1*(0.1)^1 * (0.4)^5 = 210*(0.5)^4 * 6*(0.1) * (0.4)^5 = 0.08064

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.