Sheila decides to use a standard 6-sided fair die to determine the color of her dress (blue or white) according to the following model:
- If she wore a blue dress yesterday and she rolls a four, she will wear a white dress today. Otherwise, she will wear a blue dress.
- If she wore a white dress yesterday and she rolls a four or less, she will wear a white dress again today. Otherwise, she will wear a blue dress.
Let B be the event that she wears a blue dress on the nth day and W be the event that she wears a white dress on the nth day.
(a) Using the information given, state P(Wn|Bn-1), P(Bn|Bn-1), P(Wn|Wn-1), and P(Bn|Wn-1).
(b) Suppose that Sheila wears a blue dress on day 1, that is, P(B1) = 1.
i. What is the probability that she wears a blue dress on day 2 (i.e., what is P(B2))?
ii. What is the probability that she wears a white dress on day two?
iii. Find P(B3 ∩ B2).
iv. Find P(B3 ∩ W2).
v. What is the probability that Sheila wears a blue dress on day three?
vi. What is the probability that Sheila wore a blue dress on day two, given that she wore a blue dress on day three?
(13 marks)