Question

of cars and the number of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint The joint probability distribution of the number probability table. p(x, y) 0 1 2 0 0.015 0.010 0.025 1 0.030 0.0200.050 2 0.075 0.050 0.125 3 0.090 0.060 0.150 4 0.060 0.040 0.100 5 0.030 0.0200.050 (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? Plexactly one car) = Plexactly 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

from above:

	y
x	0	1	2	total
0	0.015	0.01	0.025	0.05
1	0.03	0.02	0.05	0.1
2	0.075	0.05	0.125	0.25
3	0.09	0.06	0.15	0.3
4	0.06	0.04	0.1	0.2
5	0.03	0.02	0.05	0.1
total	0.3	0.2	0.5	1

a)

P(X=1,Y=1)=

0.02

b)

P(X<=1,Y<=1)=0.015+0.01+0.03+0.02 =

0.075

c)

P(exactly one car)=P(X=1)=			0.1
P(exactly one bus)=P(Y=1)=			0.2

d)

P(X+Y>5) =

0.62

e)

Yes. X and Y are independent as P(x,y) = P(x)*P(Y)