Advika chooses one of three scarves every morning: Red , Blue , or Green . • The probability she chooses Red is 20%. • The probability she chooses Blue is twice the probability of choosing Red. • On the remaining days she wears a Green scarf. Once a scarf is chosen, she decides whether to wear a Hat (H) and Sunglasses (S) . These choices are independent of each other but depend on the scarf colour: Scarf colour P(H)

Step-by-step solution

Let R , B , and G be the events of choosing Red, Blue, and Green scarves respectively. P(R) = 0.2 P(B) = 2 0.2 = 0.4 P(G) = 1 - (0.2 + 0.4) = 0.4 Let E be the event of wearing both a Hat and Sunglasses. Since the choices are independent for a given scarf: P(E|R) = P(H|R) P(S|R) = 0.5 0.8 = 0.4 P(E|B) = P(H|B) P(S|B) = 0.4 0.5 = 0.2 P(E|G) = P(H|G) P(S|G) = 0.1 0.5 = 0.05 Using Bayes' theorem, the required probability is: P(R|E) = P(E|R)P(R) P(E|R)P(R) + P(E|B)P(B) + P(E|G)P(G) P(R|E) = 0.4 0.2 0.4 0.2 + 0.2 0.4 + 0