The joint probability distribution of the number X of cars and the number Y of b

Question

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. p(x, y) 0 0.010 0.015 0.025 1 0.020 0.030 0.050 20.050 0.0750.125 3 0.060 0.090 0.150 4 0.040 0.060 0.100 5 0.020 0.030 0.05 (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?

Explanation / Answer

a) P(Exactly one car and exactly one bus) = 0.030

b) P(At most one car and at most one bus) = 0.010 + 0.015 + 0.020 + 0.030 = 0.075

c) P(Exactly one car) = 0.020 + 0.030 + 0.050 = 0.10

P(Exactly one bus) = 0.015 + 0.030 + 0.075 + 0.090 + 0.060 + 0.030 = 0.30

d) P(Overflow)

= 1 - P(Not overflow)

= 1 - [P(0 bus and upto 5 cars) + P(1 bus and upto 2 cars)]

= 1 - (0.010 + 0.020 + 0.050 + 0.060 + 0.040 + 0.020 + 0.015 + 0.030)

= 0.755

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