HOMEWORK 4PSTAT 120A – W17Professors Hohn & Wildman1. Suppose we draw 3 marbles from a jar containing 5 red and 7 blue marbles. At each stage, aball is drawn and its color noted. Then the marble is returned to the jar along with anothermarble of the same color. Given that the first marble is red, what is the probability that thenext two are blue?2. (adapted from Ross, 4.1) Two candies are chosen randomly from a bag of Valentine’s Day candycontaining 8 Snickers (S), 4 Peanut Butter Cups (P), and 2 Baby Ruths (B). Suppose that wewin $2 for each S and we lose $1 for each P selected. Let X denote our winnings.(a) What is the state space of X?(b) What is the probability mass function of X?3. (Ross, 4.19) Suppose that the distribution function of X is given byFX(b) =0 b < 01/2 0 ≤ b < 13/5 1 ≤ b < 24/5 2 ≤ b < 39/10 3 ≤ b < 3.51 b ≥ 3.5(a) What is the state space of X?(b) Calculate the probability mass function of X.4. Naruto is taking an important multiple choice ninja exam that is 5 questions long. Eachquestion has 3 possible answers. What is the probability that Naruto will get 4 or more correctanswers just by guessing? (If X is the (random variable giving the) number of correct answersNaruto gets while guessing, then convince yourself that Xd= Bin(n, p) for a good choice of nand a good choice of p.)5. (adapted from Ross, 4.60) Suppose that the number of times that Vampire Bill gets attacked byHep V in a given year is a Poisson random variable with parameter λ = 5. Now, he discovereda new drug (based on large quantities of vitamin C) that has just been marketed to protectfrom Hep V. It claims to reduce the Poisson parameter to λ = 3 for 75% of the population.For the rest of the population, the drug has no appreciable effect on Hep V. If Bill tries thedrug for a year and only had 2 Hep V attacks in that time, how likely is it that the drug wasbeneficial for him?6. Suppose apples are rotten with probability 0.01, independently of one another. Raider Joessells the apples in bags of 10 and offers a money-back guarantee that at most 1 of the 10 applesin the bag will be rotten. The guarantee is that a customer can return an entire bag of apples ifhe or she finds more than 1 rotten apple in it. If someone buys 3 bags, what is the probabilitythat he or she will return exactly 1 of the bags?7. (adapted from Ross, 4.72) The Cubs and Blue Jays play in the World Series where the firstteam to win 4 games is declared the overall winner. Suppose that the Cubs are stronger thanthe Blue Jays, and the Cubs win each game with probability 0.6, independently of the outcomesof the other games.(a) Find the probability, for i = 4, 5, 6, 7, that the Cubs win the World Series in exactly igames.(b) What is the probability that the Cubs would win in a 2-out-of-3 series instead?8. (Ross, 4.78) A jar contains 4 white and 4 black marbles. We randomly choose 4 marbles. If 2of them are white and 2 are black, we stop. If not, we replace the marbles in the jar and againrandomly select 4 marbles. This continues until exactly 2 of the 4 chosen are white. What isthe probability that we shall make exactly n selections?9. Suppose that the number of Bigfoot sightings per year in the Northwestern US is well-modeledby a Poisson random variable with an average of 3 sightings occurring per year. Calculate theprobability that in a given year there are at least 4 sightings in this region, given that thereare at least 2 sightings.10. Let Xd= Bin(5, 1/4).(a) Find the probability mass function of the random variable Y = X − 2.(b) What is P(X2+ X ≥ 1)?Hint: Note that you are trying to find P(X2+ X − 1 ≥ 0). One way to solve this problemis to let Z = X2+ X − 1 and find the probability mass function of Z. Then, find theprobability that P(Z ≥ 0).11. (Ross, 4.13, #tbt) A door-to-door salesman has scheduled two appointments to sell encyclo-pedias. His first appointment will lead to a sale with probability 0.3, and his second will leadindependently to a sale with probability 0.6. Any sale made is equally likely to be either forthe deluxe model, which costs $1000, or the standard model, which costs $500. Determine theprobability mass function of X, where X is the total dollar value of all sales.Page
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