Question

5.1) The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.

y
p(x, y)		0	1	2
x	0	0.010	0.015	0.025
	1	0.020	0.030	0.050
	2	0.050	0.075	0.125
	3	0.060	0.090	0.150
	4	0.040	0.060	0.100
	5	0.020	0.030	0.050

(a) What is the probability that there is exactly one car and exactly one bus during a cycle?

(b) What is the probability that there is at most one car and at most one bus during a cycle?

(c) What is the probability that there is exactly one car during a cycle? Exactly one bus?

P(exactly one car) =

P(exactly one bus) =

(d) Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle?

Answer 1

a) P(exactly one car and exactly one bus during a cycle) = 0.030

b) P(at most one car and at most one bus during a cycle) = P(0 car 0 bus) + P(0 car 1 bus) + P(1 car 0 bus) + P(1 car 1 bus)

= 0.010 + 0.015 + 0.020 + 0.030

= 0.075

c) P(exactly one car) = 0.020 + 0.030 + 0.050

= 0.100

c) P(exacly one bus) = 0.015 + 0.030 + 0.075 + 0.090 + 0.060 + 0.030

= 0.300

d) P(no overflow during a cycle) = P(upto 5 cars and 0 buses) + P(upto 2 cars and 1 bus)

= 0.010 + 0.020 + 0.050 + 0.060 + 0.040 + 0.020 + 0.015 + 0.030 + 0.075

= 0.320

P(overflow) = 1 - 0.320

= 0.680