Question

Suppose that Haley encounter two traffic lights on her commute to GGC campus. Based on

her past experience, she judge that the probability is 0.35 that the first light will be red when she gets to it,

0.45 that the second light will be red, and 0.20 that both lights will be red. Let R1 denote the event that the

first light is red, and let R2 denote the event that the second light is red.

a) Determine the probability that at least one light will be red. (Show how you calculate this.)

b) Determine the conditional probability that the second light will be red given that the first light is

red. (Show how you calculate this.)

c) Are the events R1 and R2 independent? Justify your answer numerically.

d) Are the events R1 and R2 disjoint (mutually exclusive)? Justify your answer numerically.

Answer 1

Answer:

Explained below.

Step-by-step explanation:

The information provided is,

P (R₁) = 0.35

P (R₂) = 0.45

P (R₁ ∩ R₂) = 0.20

(a)

Compute the probability that at least one light will be red as follows:

P (at least one light will be red) = P (R₁ ∪ R₂)

                                                    = P (R₁) + P (R₂) - P (R₁ ∩ R₂)

                                                    = 0.35 + 0.45 - 0.20

                                                    = 0.60

Thus, the probability that at least one light will be red is 0.60.

(b)

Compute the conditional probability that the second light will be red given that the first light is red as follows:

$P(R_{2}|R_{1})=\frac{P(R_{1}\cap R_{2})}{P(R_{1})}$

                $=\frac{0.20}{0.35}\\\\=0.571428\\\\\approx 0.57$

Thus, the conditional probability that the second light will be red given that the first light is red is 0.57.

(c)

If two events, say A and B are independent then:

P (A|B) = P (A).

The value of P (R₂ | R₁) = 0.57.

And the value of P (R₂) = 0.45

P (R₂ | R₁) ≠ P (R₂)

Thus, the events R₁ and R₂ are not independent.

(d)

If two events, say A and B are mutually exclusive then:

P (A ∩ B) = 0

The value of P (R₁ ∩ R₂) = 0.20.

Thus, the events R₁ and R₂ are not mutually exclusive.