1. STAT 240 PROBLEM SOLVING SESSION #1

2. Example: Random Variables A student’s state from this course at the end of the semester defines a random variable, S.  Values:{Pass,Fail}  Probabilities:  There are multiple ways of assigning probabilities. You need some assumptions. First is that these probabilities are assigned for an average student. Second, students are equivalent in all respects (iq, motivation,…etc) 1.They might be already given. Say Pr(S=pass)=0.7 2.You are totally ignorant. Then Pr(S=pass)=0.5 3.You have some data. You have heard that 30 students among 40 have passed from this course previous semester. So you consider nothing has changed. Then Pr(S=pass)=0.75

3. Example cts(II) One way or another you have identified your random variable. SPr(S) Pass0.6 Fail0.4 With this assignment let’s see what we can do. Consider 2 friends. They want to know whether all can pass from the course. State of each student at the end of the semester is independent of each other. Let’s try to answer this problem. A concept we will talk about a lot!

4. Example cts(III) – basic intuition behind multiplication rule. Now consider Student 1. The box below defines the probabilities of the state of the student at the end of the semester. Pass: 60% Fail: 40% Student 1’s state Student 2’s state for each state of Student 1 The size of this rectangle is 40%. Of this 40%, with 60% probability student 2 passes. So green area coresponds to p(Student1=fail, Student2=pass)=0.4x0.6=0.24 Similarly lets find, p(Student1=fail, Student2=fail)= p(Student1=pass, Student2=pass)= p(Student1=pass, Student2=fail)= Also find, p(Student2=pass) by summing up green areas.

5. Example 2 Now that we now multiplication rule lets try to answer the same problem when there are 3 students.

6. Solving Example 1 with Binomial Formula

7. Setup the example:

8. Example 2 Using the previous setup solve 1.Find the probability that one student passes from a group of 5 students

9. Find the probability that at least one student fails from a group of 5 students.

10. Find the probability that three or more student passes from a group of 5 students.

12. Calculate the probability of observing no 4’s in 5 dice rolls.

13. Calculate the probability that the sum of a pair of dices in 3 of the 5 rolls greater than nine.