STATS 210: PROBLEM SET 4 VERSION B (MUST BE COMPLETED ON YOUR OWN) DUE: 10-14 1. Consider tossing a fair die two times. Consider the following events: A = 3 or less on the first roll, B = sum of the two rolls is at least 10. Events A and B _____. Events A and B_____. a. are disjoint, are independent b. are disjoint, are not independent c. are not disjoint, are independent d. are not disjoint, are not independent 2. Consider tossing a fair die two times. Consider the following events: A = 4 or less on the first roll, B = sum of the two rolls is equal to 7. Events A and B _____. Events A and B_____. a. are disjoint, are independent b. are disjoint, are not independent c. are not disjoint, are independent d. are not disjoint, are not independent 3. This problem involves two distinct sets of events, which we label A1and A2 and B1 and B2. The events A1 and A2 are mutually exclusive and collectively exhaustive within their set. The events B1and B2 are mutually exclusive and collectively exhaustive within their set. Intersections can occur between all events from the two sets. Given P(A1) = 0.5, P(B1|A1) = 0.4, and P(B1|A2) = 0.7, what is P(A1|B2)? (Round your answer to four decimal places.) 4. Consider the vote distribution below: Pawnee Eagleton Harry 0.8 p Voldemort 0.2 1-p The distribution above represents the results of 100 votes from Pawnee citizens and 200 votes from Eagleton citizens. Find a value of p such that the probability that a randomly selected vote for Voldemort (from those collected) belongs to an Eagletonian is equal to 0.9. 5. You are convinced that the rain depends on whether or not you bring an umbrella. To investigate, you keep track of days when it rains and days when you bring an umbrella. You find that you brought your umbrella 25% of days, and that when you brought your umbrella, it rained on 10% of days. When you didn't bring your umbrella, it rained 20% of days. If it's raining today, what's the probability you brought an umbrella