REVISION SESSIONAL TEST

1. REVISIONSESSIONAL TEST

2. Variables • Qualitative variables take on values that are names or labels. The colour of a ball (e.g., red, green, blue) • Quantitative variables are numeric. They represent a measurable quantity. For example, population of a city

3. Discrete vs Continuous Variables Quantitative variables can be further classified as discrete or continuous. If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable.

4. Basic Concepts • Consider following set of values: 12, 15, 21, 27, 20, 21 • Mean • Mode • Median • Variance • Standard Deviation • Range

5. Basic Concepts • Consider following set of values: 12, 15, 21, 27, 20, 21 • Stem-and-Leaf Plot • Box-and-Whisker Plot • Frequency Distribution Table • Histogram

6. Trial • A single performance of an experiment is called a Trial. • The result of a trial is called an Outcome or a Sample Point. • The set of all possible outcomes of an experiment is called a Sample Space. • Subsets of a Sample Space are called Events.

7. Examples – Sample Space • The set of integers between 1 and 50 and divisible by 8 • S = { x! x2 + 4x – 5 = 0 } • Set of outcomes when a coin is tossed until a tail or three heads appear • Set of sampling items randomly until one defective item is observed.

8. Definitions • An event is a subset of a Sample Space. • The complementof an event A with respect to S is the subset of all elements of S that are not in A. • The intersection of two events A and B is the event containing all elements that are common to A and B. • The union of two events A and B is the event containing all the elements that belong to A or B or both.

9. Mutually Exclusive Events • Two events A and B or Mutually Exclusive or Disjoint, if A and B have no elements in common.

10. Probability If the Sample Space S of an experiment consists of finitely many outcomes (points) that are equally likely, then the probability P(A) of an event A is P(A) = Number of Outcomes (points) in A Number of Outcomes (points) in S

11. Permutation • A permutation is an arrangement of all or part of a set of objects. • Number of permutations of n objects isn! • Number of permutations of n distinct objects taken r at a time is nPr = n! (n – r)! • Number of permutations of n objects arranged is a circle is (n-1)!

12. Permutations • The number of distinct permutations of n things of which n1 are of one kind, n2 of a second kind, …, nk of kth kind is n! n1! n2! n3! … nk!

13. Problem • How many permutations of 3 different digits are there, chosen from the ten digits, 0 to 9 inclusive? A 84 B 120 C 504 D 720

15. Problem • In how many ways can a Committee of 5 can be chosen from 10 people? A 252 B 2,002 C 30,240 D 100,000

16. Independent Probability • If two events, A and B areindependentthen theJoint Probabilityis P(A and B) = P (A Π B) = P(A) P(B) • For example, if two coins are flipped the chance of both being heads is  1/2 x 1/2 = 1/4

17. Mutually Exclusive • If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as  P (A U B). • If two events are Mutually Exclusivethen the probability of either occurring is P(A or B) = P (A U B) = P(A) + P(B) • For example, the chance of rolling a 1 or 2 on a six-sided die is  1/6 + 1/6 = 2/3

18. Conditional Probability • Conditional Probabilityis the probability of some event A, given the occurrence of some other event B. • Conditional probability is written as P(A І B), and is read "the probability of A, given B". It is defined by P(A І B) = P (A Π B) P(B)

19. Conditional Probability • Consider the experiment of rolling a dice. Let A be the event of getting an odd number, B is the event getting at least 5. Find the Conditional Probability P(A І B).

20. Independent Events Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. P(A and B) = P(A) · P(B)

21. Complementation Rule For an event A and its complement A’ in a Sample Space S, is P(A’) = 1 – P(A)

22. Example - Complementation Rule 5 coins are tossed. What is the probability that: • At least one head turns up • No head turns up

23. Problem List following of vowel letters taken 2 at a time: • All Permutations • All Combinations without repetitions • All Combinations with repetitions

24. Probability Mass Function A probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.

25. x P(x≤A) 1 P(x≤1)=1/6 2 P(x≤2)=2/6 3 P(x≤3)=3/6 4 P(x≤4)=4/6 5 P(x≤5)=5/6 6 P(x≤6)=6/6 Cumulative Distribution Function

26. Probability Density Function A probability density function (pdf) describes the relative likelihood for the random variable to take on a given value.

27. Discrete Distribution: Formulas • P(a < X ≤ b) = F(b) – F(a) = (Sum of all Probabilities)

28. Continuous Distribution: Formulas • P(a < X ≤ b) = F(b) – F(a) = dv dv = 1 (Sum of all Probabilities)

29. Uniform Distribution • f(x) = 1/b-a a ≤ x ≤ b 0 Otherwise • F(x) = x-a/b-a a ≤ x ≤ b 0 Otherwise

30. Review Question Two dice are rolled and the sum of the face values is six? What is the probability that at least one of the dice came up a 3? • 1/5 • 2/3 • 1/2 • 5/6 • 1.0

31. Review Question Two dice are rolled and the sum of the face values is six. What is the probability that at least one of the dice came up a 3? • 1/5 • 2/3 • 1/2 • 5/6 • 1.0 How can you get a 6 on two dice? 1-5, 5-1, 2-4, 4-2, 3-3 One of these five has a 3. 1/5