PS - Probability and Set Notation Lesson

Probability and Set Notation

At the heart of probability involving sets are the ideas of  intersections  and  unions . It is important to understand the difference between the two.  

Intersection: Imagine you are standing in the intersection of two roads. "A Street" and "B Avenue". Which road are you on? Are you on "A Street"? Are you on "B Avenue"? You may have thought, "I'm on both at the same time" and you would be right.

An intersection of two sets are the elements that are contained in both sets at the same time. They are where the two sets intersect or overlap. The symbol for intersection is $\cap$. If you want to say "The intersection of sets A and B", you write  $A\cap B$ .

Union: The United States is a union of individual states. If you are in Georgia, are you in the United States? What if you are in Florida, are you still in the United States? Do you have to be in both Georgia AND Florida to be in the United States? Of course not.

A union of two sets are all of the elements that are in one set or the other or both, everything that is in either set is in the union. The symbol for union is  $\cup$.  If you want to say the union of sets A and B, you write  $A\cup B$ .

Look below for a summary of symbols that will be used in this unit and their meanings:

Symbol	Meaning	Details
$\cap$	Intersection	The intersection of two (or more sets) are all the elements that appear in both (or all) sets.
$\cup$	Union	The union of two sets is everything in both sets.
A	Set A	All of the elements in set A. They can be in set A exclusively, or as part of an intersection with another set.
$A'\:or\:\overline{A}$	Not Set A	"Not Set A" includes any elements that are not in set A. Any element(s) in set A (including elements that intersect) is (are) not a part of "Not Set A."
P(A)	Probability of Outcome Being in Set A	This is equal to the number of outcomes in set A divided by the total number of outcomes.
$P\left(A'\right)$	Probability of Outcome Not Being in Set A	This is equal to the number of outcomes not in A divided by the total number of outcomes. It can also be found by using the following: 1 - P(A).
$P\left(A|B\right)$	Probability of A given B	This is asking for the probability of A occurring given that B has already occurred.

Basic Probability

As you have learned in past courses, the probability of an outcome is the number of successes divided by the number of possible outcomes. This is usually written as a fraction, a decimal, or a percent.

Example 1: The Classic Marble Problem

In a bag are 3 Red marbles, 4 Blue marbles, 2 Yellow marbles and 1 Green marble.   If you draw out a marble at random, what is the probability that you will draw out a blue marble?

$P\left(Blue\right)=\frac{Total ofBlue}{Total of Marbles}=\frac{4}{10}=\frac{2}{5}=.40\:or\:40\%$$

 

Example 2: And the Winner Is...

A school is awarding a $100 prize to one student who attended all the SAT prep sessions that the school offers. Eight students have qualified for the drawing. Three are boys: Brian, Rich and Adam. Five are girls: Lisa, Rhonda, Betsy, Jill, and Allison. If they choose a student at random, find the following probabilities:

a. The probability they choose a girl:  $P(Girl)=\frac{Number\:of\:girls}{Total\:number\:of\:students}=\frac{5}{8}=.625\:or\:62.5\%$  

b. The probability they choose a student whose name begins with "A":  $P\left(A\:name\right)=\frac{Number\:of\:Names\:that\:Start\:with\:A}{Total\:number\:of\:students}=\frac{2}{8}=\frac{1}{4}=.25\:or\:25\%$ 

c. The probability they choose a girl whose name begins with "A":  $P\left(Girl\:\cap\:A\right)=\frac{Number\:of\:Girls\:with\:Names\:That\:Start\:with\:A}{Total\:Number\:of\:Students}=\frac{1}{8}=.125\:or\:12.5\%$

P(Consonant) = probability of the student whose first letter of their name is a consonant. P(Vowel)=means probability of the Student whose first letter of their name is a Vowel.

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