22.4UC2 Tutorial Exercises — Week 3

Pick some of the exercises below for your tutorial.

Exercise 1 Recall from calculus that (a + b)^{n} = ∑_{i=0}^{n} \binom{n}{k}a^{k}b^{n−k}.

(a). Use this fact to show that for a binomial probability distribution of length n and success probability p, ∑_{all x} p(x) = 1.

(b). Prove that if x has binomial probability distribution with parameters n and p then E(x) = np.

[Hint: Write out the definition of E(x), factor out n and p, to get E(x) = np ∑ …, and recognise the remaining sum as the probabilities of a Binomial random variable for some n′ and p′.]

Exercise 2 On New Year’s Eve the police makes random controls of car drivers. Suppose they find that about 20% of the drivers have consumed alcohol. What is the probability that among five checked drivers at most 3 have drunk? [99.33%.]

Exercise 3 Say the probability of passing the road test (for a driver’s licence) is 75%, without changing with every try. What are the probabilities that a young drier will pass the test on the second (or fourth) attempt? [18.75%, 1.17%.]

Exercise 4 Among 100 applicants to university only 60 are actually qualified. If a university selects randomly 6 students for interviews, what is the probability that only 3 of them are qualified? (Do two calculations, using the binomial and the hyper-geometric distribution.) [27.65%, 28.36%.]

Exercise 5 Burning tiles is a difficult task. Suppose that certain tiles are known to have 5 defects per 10 square feet (with Poisson distribution). What is the probability that a randomly selected tile of 1 square foot is of high quality, i.e., has no defect? [60%.]

Exercise 6 What is the probability that in a room of x people at least two of them are born on the same day? How many people are needed so that this probability is greater than 50%? What is the probability that among 12 people 2 have their birthday on the same day? [1 − \frac{365·364···(365−x+1)}{365^{x}}, 23 (then the probability is 50.73%), 16.70%.]

Exercise 7 A regional airport has about 56.4 takeoffs per day. Suppose that the random variable takeoffs has a Poisson distribution. What is the probability that there will be at least 2 takeoffs within one hour? What is the probability that there is at least 1 hour between two takeoffs? How do these values change if takeoffs happen only between 8 am and 8 pm? (Frankfurt airport has about 1200 takeoffs per day.) [68.05%, 65.34%, 94.82%, 80.83%.]