Counting and Probability

Counting Elements of Sets Theorem. The Inclusion/Exclusion Rule for Two or Three Sets If A, B, and C are finite sets, then N(A  B) = N(A) + N(B) – N(A  B) and N(A  B  C) = N(A) + N(B) + N(C) – N(A  B) – N(A  C) – N(B  C) + N(A  B  C)

Example 1 In a class of 50 college freshmen, 30 are studying BASIC, 25 studying PASCAL, and 10 are studying both. How many freshmen are studying either computer language? A: set of freshmen study BASIC B: set of freshmen study PASCAL N(A  B) = N(A)+N(B)-N(A  B) = – 10 =

Example 2 A professor takes a survey to determine how many students know certain computer languages. The finding is that out of a total of 50 students in the class, 30 know Java; 18 know C++ 26 know SQL 9 know both Java and C++ 16 know both Java and SQL 8 know both C++ and SQL 47 know at least one of the 3 languages.

Example 2 a. How many students know none of the three languages? b. How many students know all three languages? c. How many students know Java and C++ but not SQL? How many students know Java but neither C++ nor SQL Answer: a. 50 – 47 = 3

Example 2 J = the set of students who know Java C = the set of students who know C++ S = the set of students who know SQL Use Inclusion/Exclusion rule.

Discrete Probability The probability of an event is the likelihood that event will occur. “ Probability 1 ” means that it must happen while probability 0 means that it cannot happen E.g.: the probability of … “ Manchester United defeat Liverpool this season” is 1 “ Liverpool win the Premier League this season ” is 0 Events which may or may not occur are assigned a number between 0 and 1.

Discrete Probability Consider the following problems: What ’ s the probability of tossing a coin 3 times and getting all heads or all tails? What ’ s the probability that a list consisting of n distinct numbers will not be sorted? Set cardinalities are useful.

Discrete Probability An experiment is a process that yields an outcome. A sample space is the set of all possible outcomes of a random process. An event is an outcome or combination of outcomes from an experiment. An event is a subset of a sample space. Examples of experiments: - Rolling a six-sided die - Tossing a coin.

Example EG: What ’ s the probability of tossing a coin 3 times and getting all heads or all tails? Can consider set of ways of tossing coin 3 times: S = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} S is called a sample space. Next, consider set of ways of tossing all heads or all tails: E = {HHH,TTT} E is called the event. Assuming all outcomes equally likely to occur Definition: The probability of the event E is the ratio p (E ) = |E | / |S | EG: Our case: p (E ) = 2/8 = 0.25

Example Five microprocessors are randomly selected from a lot of 1000 microprocessors among which 20 are defective. Find the probability of obtaining no defective microprocessors. There are C(1000,5) ways to select 5 micros. There are C(980,5) ways to select 5 good micros. The prob. of obtaining no defective micros is C(980,5)/C(1000,5) = 0.904

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Probability of combinations of events Theorem. Let E 1 and E 2 be events in the sample space S. Then P(E 1  E 2 ) = P(E 1 ) + P(E 2 ) – P(E 1  E 2 ) E.g. What is the prob. that a positive integer selected at random from the set of positive integers not greater than 100 is divisible by either 2 or 5. E 1 : event that the integer selected is divisible by 2 E 2 : event that the integer selected is divisible by 5 P(E 1  E 2 ) = 50/ /100 – 10/100 = 3/5

Exercise Two fair dice are rolled. What is the probability that the sum of the numbers on the dice is 10?

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