3 Practical Confidence Intervals (a) It's New Year's Eve, and you're re-evaluating your finances for the next year. Based on previ- ous spending patterns, you know that you spend $1500 per month on average, with a standard deviation of $500, and each month's expenditure is independently and identically distributed. As a college student, you also don't have any income. How much should you have in your bank account if you don't want to run out of money this year, with probability at least 95%? (b) As a UC Berkeley CS student, you're always thinking about ways to become the next billion- aire in Silicon Valley. After hours of brainstorming, you've finally cut your list of ideas down to 10, all of which you want to implement at the same time. A venture capitalist has agreed to back all 10 ideas, as long as your net return from implementing the ideas is positive with at least 95% probability. Suppose that implementing an idea requires 50 thousand dollars, and your start-up then suc- ceeds with probability p, generating 150 thousand dollars in revenue (for a net gain of 100 thousand dollars), or fails with probability 1- p (for a net loss of 50 thousand dollars). The success of each idea is independent of every other. What is the condition on p that you need to satisfy to secure the venture capitalist's funding? (c) One of your start-ups uses error-correcting codes, which can recover the original message as long as at least 1000 packets are received (not erased). Each packet gets erased independently with probability 0.8. How many packets should you send such that you can recover the mes- sage with probability at least 99%? 4 Estimating x In this problem, we discuss some interesting ways that you could probabilistically estimate n, and see how good these techniques are at estimating It. Technique 1: Buffon's needle is a method that can be used to estimate the value of n. There is a table with infinitely many parallel lines spaced a distance I apart, and a needle of length I. It turns out that if the needle is dropped uniformly at random onto the table, the probability of the needle intersecting a line is =. We will see a proof of this later on in the notes. Technique 2: Consider a square dartboard, and a circular target drawn inscribed in the square dartboard. A dart is thrown uniformly at random in the square. The probability the dart lies in the circle is 7