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? Lets number the cards from 1 through 13. There is a finite set of the couples that are “next to each other”: {1,2},{2,3},{3,4}….. How many in total? Apparently, the last couple is {12,13}, and there is 12 couples in total. In addition, each couple can be composed in two ways: {2,3} an {3,2}. Thus, the total number of selections satisfying the condition is 2*12 = 24. The total number of all possible selections is Binomial[12,2]. Thus, P=24/Binomial[12,2]=4/11.

6. 6 TEST 1.Put the name on the top of the page. 2.Solve the problems on the separate sheets of paper. 3.Provide the detailed solutions explaining your calculations. 4.At the end of each solution, add “Answer: ”. 5.Staple the solutions to the first page and submit it in 55 minutes. Good Luck!

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