Answer to Question #340127 in Statistics and Probability for Maneo

Question

A commuter passes through three traffic lights on the way to work. Each light is either red, yellow, or green. An experiment consists of observing the colour of the three lights.

• i) List the 27 possible outcomes in the sample space.

• ii) Let A be the event that all the colours are the same. List the outcomes of A.

• iii) Let B be the event that all the colours are different. List the outcomes of B.

• iv) Let C be the event that at least two lights are green. List the outcomes of C.

• v) List the outcomes in 𝐴 ⋂ 𝐶.

• vi) List the outcomes in 𝐴 ⋃ 𝐵.

• vii) List the outcomes in 𝐴 ⋂ 𝐶' .

• viii) List the outcomes in 𝐴' ⋂ 𝐶 .

• ix) Are events A and C mutually exclusive? Explain.

• x) Are events B and C mutually exclusive? Explain.

Answer 1

i)

$$S=\{RRR, RRG, RRY, RGR, RGG, RGY,$$

$$RYR, RYG, RYY, GGG, GGR, GGY,$$

$$GRG, GRR, GRY, GYG, GYR, GYY,$$

$$YYY, YYR, YYG,YRY, YRR, YRG,$$

$$YGY,YGR, YGG\}$$

ii)

$$A=\{RRR, GGG, YYY\}$$

iii)

$$B=\{RGY, RYG, GRY, GYR, YRG, YGR\}$$

iv)

$$C=\{RGG, GGG, GGR, GGY,GRG,$$

$$GYG,YGG\}$$

v)

$$A\cap C=\{GGG\}$$

vi)

$$A\cup B=\{RRR, GGG, YYY,RGY, RYG, GRY,$$

$$GYR, YRG, YGR\}$$

vii)

$$A\cap C'=\{RRR, YYY\}$$

viii)

$$A'\cap C=\{RGG, GGR, GGY,GRG,$$

$$GYG,YGG\}$$

ix)

When two events (call them $D$ and $F$ ) are Mutually Exclusive it is impossible for them to happen together:

$$P(D\cap F)=0$$

$$P(D\cup F)=P(D)+P(F)$$

Events $A$ and $C$ are not mutually exclusive

$$A\cap C=\{GGG\}$$

x)

Events $B$ and $C$ are mutually exclusive.

They cannot both occur at the same time.