Introduction to probability, Complement and Venn Diagram
Probability What can we say about probability? ◦ It is a measure of likelihood, uncertainty, possibility, … ◦ And it is a number, numeric measure
Some concepts Set: ◦ A set is a collection of distinct objects, considered as an object in its own right ◦ Example: A class of stat 225 students NBA teams Result of a football game for a team (win, lose, tie) Outcomes of rolling a die ( 1, 2, 3, 4, 5, 6)
Some concepts Experiment: ◦ A process that generates well-defined outcomes. ◦ *** You must know what could possibly happen before performing the experiment. ◦ Example: Roll a die: YES ? Throw a stone out of window: NO.
Some concepts Sample space ◦ Don’t get confused with sample and sampling in statistics. ◦ A sample space for an experiment is the set of ALL experimental outcomes. ◦ Example: Toss a coin: {Head, Tail} Take an exam or quiz: { All the possible grades } Inspection of a manufactured part: {defective, nondefective}
Some concepts Some notes on sample space: 1. It could be finite or infinite, i. e. , there could be infinitely many possibilities, but you still must know all of them. Example, choosing a point on a segment, there are infinitely many choices but you know it must be on the segment. 2. Not all the sample points in the sample space are equally likely. Example: Getting a royal flush, full house or just a pair in a poker game.
Some concepts Event: ◦ A collection of sample points. It is an outcome from an experiment that may include one or more sample points. ◦ Examples: Toss a coin and get a head. Roll a die and get a 5. Toss a coin twice and get two heads. Roll a die and get an even number.
Some concepts Complement of an event A: ◦ Notation: Ac ◦ Also an event ◦ Includes ALL sample points that are not in A. Example: A: Roll a die and get an even number, Ac : Roll a die and get an odd number.
Some concept We use P(A) to represent the probability that one event occurs. Clearly: P(A) + P(Ac)=1. A and Ac are called mutually exclusive, which means any sample point call fall in either A only or Ac only, but not both.
Exercise Some of tossed a coin twice. ◦ 1. What is the sample space: { HH, HT, TH, TT } ◦ 2. Let A = { toss the coin twice and get two heads}, then P(A)=? ◦ 3. Let B = { toss the coin twice and get at least one head}, then P(B)=? ◦ 4. What is Bc , and P(Bc )=?
Exercise We have 20 fruits in a box, 10 apples, 6 pears and 4 peaches. ◦ If A={pick a fruit from the box and get an apple}, then P(A)=? ◦ If B={pick a fruit from the box and it is NOT an apple}, then P(B)=?
Exercise 50 balls are put in one box, 25 white, 15 red and 10 green. ◦ A={ pick a ball and it is green }, P{A}=? ◦ B={pick a ball and it is colored}, P{B}=?
A graphical way of dealing with probability Venn diagram: ◦ Illustrates the concept of complement. ◦ Somewhat like a crosstabulation.
Example # 1. A class of 30 students took 2 midterms during a semester. 22 of them passed the first one and 26 of them passed the second one. If 3 students failed both, find the number of students who passed the each of the two midterms and failed the other. Answer: 21 passed both, 1 passed the first but failed the second, 5 passed the second but failed the first
Example #2. We have a box and 30 balls, 16 are white and the rest are colored; 9 are plastic and the others are made of rubber. If there are 11 white rubber balls, show the breakdown of balls by color and material. Answer: 5 white plastic; 11 white rubber; 4 colored plastic and 10 colored rubber.
Example #3. In a class of 30 students, 20 are male, 15 are white and 5 are black females. Assuming only no students of other race in the class. Find the break down of students by race and gender. Answer: 10 black male; 5 black female; 5 white female and 10 white male.
Find some probabilities In #1, what is the probability of finding a student who passed at least one midterm? ◦ 27/30 In #2, pick a ball at random, what is the probability of getting a white plastic? ◦ 5/30 In #3, pick a student from the class and what is the probability of getting a white female? ◦ 5/30
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