2. Counting and Probability

Transcription

1 2. Counting and Probability

2 2.1.1 Factorials Combinatorics Probability Theory Probability Examples

3 2.1.1 Factorials

4 Combinatorics Combinatorics is the mathematics of counting. It can be quite delicate. Two important notions for us are those of combinations and permutations. In order to define these, we need the notion of factorial and binomial coefficient.

5 Factorial! The factorial of a number is simply the product of itself with all positive integers less than it. 0! = 1 By convention,. It is possible to define the factorial for non-integers, but this quite advanced and is not part of the CLEP. When computing with factorials, it is helpful to write out the multiplication explicitly, as there are often cancellations to be made.

6 Formulas and Notations n! =n (n 1) (n 2) ! = 1 2! = 2 3! = 6 4! = 24 n k = k!(n n! k)!

7 5! Compute the following:

8 6! 5!

9 n! (n 1)!

10 0! 2!

11 6 2

12 3 3

13 7 0

14 5 3

15 2.1.2 Combinatorics

16 Counting with Factorials Factorials are useful for combinatorics, i.e. problems involving counting. Given n objects, the number of groups of size k when order doesn t matter is n n! = k k!(n k)! Given n objects, the number of groups of size k when order matters is. n! (n k)

17 How many groups of 3 from 10 are possible, if order matters?

18 If order doesn t matter?

19 How many codes of length 2 may be generated from the letters {A,B,C,D,E,D} if the letters cannot be repeated, and order matters?

20 What if repeats were allowed?

21 2.2.1 Probability Theory

22 Probability Probability in mathematics quantifies random events. It is a large subject with a rich history; we focus on a few basic ideas. We consider the probability that events occur: P(A) Our goal is to understand how to compute such probabilities. Note that all probabilities have values between 0 and 1.

23 Notation Let be any two events. A, B The union of is the event that either A, B A or B occurs. It is denoted A [ B. The intersection of is the event the that A and B A, B both occur. It is denoted. The complement of is the event that does not A occur. It is denoted A c. There are some principle that dictate how to compute these quantities. A \ B A

24 Basic Principles Union Law: P(A [ B) =P(A)+P(B) P(A \ B) Events are independent if: A, B P(A \ B) =P(A)P(B) The conditional probability of on is A B P(A B) = P(A \ B) P(B) The law of the complement states P(A c )=1 P(A)

25 2.2.1 Probability Theory

26 Probability Probability in mathematics quantifies random events. It is a large subject with a rich history; we focus on a few basic ideas. We consider the probability that events occur: P(A) Our goal is to understand how to compute such probabilities. Note that all probabilities have values between 0 and 1.

27 Notation Let be any two events. A, B The union of is the event that either A, B A or B occurs. It is denoted A [ B. The intersection of is the event the that A and B A, B both occur. It is denoted. The complement of is the event that does not A occur. It is denoted A c. There are some principle that dictate how to compute these quantities. A \ B A

28 Basic Principles Union Law: P(A [ B) =P(A)+P(B) P(A \ B) Events are independent if: A, B P(A \ B) =P(A)P(B) The conditional probability of on is A B P(A B) = P(A \ B) P(B) The law of the complement states P(A c )=1 P(A)

29 2.2.1 Probability Theory

30 Probability Probability in mathematics quantifies random events. It is a large subject with a rich history; we focus on a few basic ideas. We consider the probability that events occur: P(A) Our goal is to understand how to compute such probabilities. Note that all probabilities have values between 0 and 1.

31 Notation Let be any two events. A, B The union of is the event that either A, B A or B occurs. It is denoted A [ B. The intersection of is the event the that A and B A, B both occur. It is denoted. The complement of is the event that does not A occur. It is denoted A c. There are some principle that dictate how to compute these quantities. A \ B A

32 Basic Principles Union Law: P(A [ B) =P(A)+P(B) P(A \ B) Events are independent if: A, B P(A \ B) =P(A)P(B) The conditional probability of on is A B P(A B) = P(A \ B) P(B) The law of the complement states P(A c )=1 P(A)

33 2.2.2 Probability Examples

34 Draw one card from a full deck of 52, well-shuffled. P(diamond or Queen)

35 P(ace or King)

36 P(diamond or club)

37 Roll a die twice P(roll 1 apple 2 and roll 2 apple 3)

38 P(sum of rolls = 7)

39 P(second roll di erent from first)

40 For two events A, B, suppose P(A) = 1 2, P(B) =1 2, P(A \ B) =1 3. Are A, B independent?