Question

suppose that you encounter two traffic lights on your commute to school. Based on past experience, you judge that the probability is .60 that the first light will be red when you get to it, .50 that the second light will be red, and .40 that both lights will be red.

a)Determine the conditional probability that the second light will be red, given that the first light is red. (Here and throughout, show the details of your calculations.)

b)Are the events {first light is red} and {second light is red} independent? Justify your answer.

c) Given that at least one light is red, what is the probability that both lights are red? (Show your work.)

Answer 1

P(L1)= first light will be red

P(L2)= second light will be red

P(L1L2)=both lights will be red.

#Given:

P(L1)= 0.60 P(L2)= 0.50 P(L1L2)= 0.40

a)Determine the conditional probability that the second light will be red, given that the first light is red.

Ans:

probability that the second light will be red, given that the first light is red

ie P(L2/L1)

P(L2/L1)=P(L1L2)/P(L1)=0.40/0.60=0.67

# the conditional probability that the second light will be red, given that the first light is red. is 0.67

b)Are the events {first light is red} and {second light is red} independent?

# the two events are said to be independent if P(L1L2)=P(L1)*P(L2)

P(L1)*P(L2)=0.60*0.50=0.3

THEREFORE

hence vents {first light is red} and {second light is red} are not independent

c) Given that at least one light is red, what is the probability that both lights are red?

#the probability that both lights are red is 0.40