iva loves coffee but her nurse friend recommended reducing her coffee intake and she decided to do it by using a fair 10-sided die to determine her coffee intake according to the following simple model. If she drank coffee yesterday then she will drink any other hot drink (no-coffee) today if she rolls a seven or less, otherwise she will drink coffee. If she drank no-coffee yesterday then she will drink no-coffee today if she rolls a six or more, otherwise she will drink coffee. Let Cn be the event that she drinks coffee on the nth day and Nn be the event that she drinks no-coffee on the nth day. (a) Using the information given, write down the value of these probabilities P (Cn | Cn−1), P (Cn | Nn−1), P (Nn | Cn−1) and P (Nn | Nn−1). [2 marks] (b) Suppose Siva drinks no-coffee on day 1, that is, P(N1) = 1. i. What is the probability that Siva drinks no-coffee on day 2? [1 mark] ii. What is the probability that Siva drinks coffee on day 2? [1 mark] iii. Find P(C3 ∩ C2) and P(C3 ∩ N2). [2 marks] iv. What is the probability that Siva drinks coffee on day 3? [1 mark] v. Suppose Siva drinks coffee on day 3. What is the probability that she drank no-coffee on day 2?
Question
iva loves coffee but her nurse friend recommended reducing her coffee intake and she decided to do it by using a fair 10-sided die to determine her coffee intake according to the following simple model. If she drank coffee yesterday then she will drink any other hot drink (no-coffee) today if she rolls a seven or less, otherwise she will drink coffee. If she drank no-coffee yesterday then she will drink no-coffee today if she rolls a six or more, otherwise she will drink coffee. Let Cn be the event that she drinks coffee on the nth day and Nn be the event that she drinks no-coffee on the nth day. (a) Using the information given, write down the value of these probabilities P (Cn | Cn−1), P (Cn | Nn−1), P (Nn | Cn−1) and P (Nn | Nn−1). [2 marks] (b) Suppose Siva drinks no-coffee on day 1, that is, P(N1) = 1. i. What is the probability that Siva drinks no-coffee on day 2? [1 mark] ii. What is the probability that Siva drinks coffee on day 2? [1 mark] iii. Find P(C3 ∩ C2) and P(C3 ∩ N2). [2 marks] iv. What is the probability that Siva drinks coffee on day 3? [1 mark] v. Suppose Siva drinks coffee on day 3. What is the probability that she drank no-coffee on day 2?
Solution
(a) The probabilities are as follows:
P(Cn | Cn−1) = 3/10, because if she drank coffee yesterday, she will drink coffee today if she rolls an 8, 9, or 10 on a 10-sided die, which is a 3 in 10 chance.
P(Cn | Nn−1) = 5/10, because if she drank no-coffee yesterday, she will drink coffee today if she rolls a 1, 2, 3, 4, or 5 on a 10-sided die, which is a 5 in 10 chance.
P(Nn | Cn−1) = 7/10, because if she drank coffee yesterday, she will drink no-coffee today if she rolls a 1, 2, 3, 4, 5, 6, or 7 on a 10-sided die, which is a 7 in 10 chance.
P(Nn | Nn−1) = 5/10, because if she drank no-coffee yesterday, she will drink no-coffee today if she rolls a 6, 7, 8, 9, or 10 on a 10-sided die, which is a 5 in 10 chance.
(b) i. The probability that Siva drinks no-coffee on day 2 is P(N2 | N1) = 5/10.
ii. The probability that Siva drinks coffee on day 2 is P(C2 | N1) = 5/10.
iii. P(C3 ∩ C2) = P(C3 | C2) * P(C2) = (3/10) * (5/10) = 15/100.
P(C3 ∩ N2) = P(C3 | N2) * P(N2) = (5/10) * (5/10) = 25/100.
iv. The probability that Siva drinks coffee on day 3 is P(C3) = P(C3 ∩ C2) + P(C3 ∩ N2) = 15/100 + 25/100 = 40/100 = 4/10.
v. If Siva drinks coffee on day 3, the probability that she drank no-coffee on day 2 is P(N2 | C3) = P(C3 ∩ N2) / P(C3) = (25/100) / (4/10) = 25/40 = 5/8.
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