1. 1 Lecture 4: Warm-up Test. Discussion of the test problems Experiments with Random Numbers

2. 2 Warming up Compare two problems: 1. There are 20 eggs in the box, 4 of them are rotten. You randomly chose three eggs. What it is the probability that (a)All three are rotten (b)Two of them are rotten. 2. One of five apples in the orchard is rotten. You randomly pick 3 apples. What it is the probability that (a)All three are rotten (b)Two of them are rotten. Which of two is described by the regular Binomial distribution? Why?

3. 3 1. There are 20 eggs in the box, 4 of them are rotten. You randomly chose three eggs. What it is the probability that (a)All three are rotten (b)Two of them are rotten. Solution: (a)P(3) = N( out of 3 )/N(total) = C 4,3 /C 20,3 ~0.004 (b)P(2)=C 4,2 C 16,1 /C 20,3 ~0.084 2. One of five apples in the orchard is rotten. You randomly pick 3 apples. What it is the probability that All three are rotten Two of them are rotten. In this case we are dealing with classical Bernoulli trials: the possible outcomes in all experiments (trials) are described by the probabilities p and q=1-p. Solution: P(3)= C 3,3 (1/5) 3 =0.008 P(2) = C 3,2 (1/5) 2 (4/5) 1 ~0.1

4. 4 You randomly pull 2 out of 13 spades. What is the probability that their ranks are next to each other?

5. 5 You randomly pull 2 out of 13 spades. What is the probability that their ranks are next to each other? Consider n cards. There are n-1 couples that are the next neighbors: {1,2},{2,3},…., {n-1,n} (we ignore the order). What is the total number of ways of selecting a couple (if order does not matter): N total = Binomial[n,2]. The probability equals: P (neighbors)= (n-1) 2 (n-2)!/n! = 2/n. The answer is 2/13.

6. 6 TEST 1.Put the name on the top of the page. 2.Solve the problems. 3.Provide the detailed solutions explaining your calculations. 4.At the end of each solution, add “Answer: ”. 5.Submit the solutions in 65 minutes. Good Luck!

7. 7 Solutions of the test problems In a separate file