University of California Berkeley

### Question Description

I’m working on a Statistics question and need guidance to help me study.

Question 1.

An airplane is missing. Based on its flight plan, it has been determined that the airplane is equally likely to be in any one of three locations. A search team will find the airplane in the ith location (assuming it is there) with probability pi = 1 − i/(i + 1), for i = 1, 2, 3. Find the conditional probability that the airplane is in the second (i = 2) location given that the airplane was not found while searching the first (i = 1) location

Question2.

An urn initially contains r red balls and b blue balls. In each step, a ball is chosen uniformly at random, and then put back into the urn together with a new ball of the same color. Let Ri be the event that in step i a red ball is chosen from the urn. Show that P(R1 ∩ R2) = P(R2 ∩ R3).

Question 3.

A coin has two sides, Heads and Tails. When flipped it comes up Heads with an unknown probability p and Tails with probability q = 1−p. Let ˆp be the proportion of times it comes up Heads after n flips. Using Normal approximation, find n so that |p−pˆ| ≤ 0.01 with probability approximately 95% (regardless of the actual value of p). You may use the following facts: Φ(−2, 2) = 95% pq ≤ 1/4 for any p ∈ [0, 1].