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MATH 2600 Probability and Statistics Chapter 4 notes
Section 4-2 Fundamentals Key Concept This section introduces the basic concept of the ______________________________________. Different methods for finding probability values will be presented. The most important objective of this section is to learn how to _______________________ probability values.
Event – ___________________________________________________________________________
Simple Event – ___________________________________________________________________________
Sample space – __________________________________________________________________________
Experiment or procedure An example of a Simple event Sample Space
Roll a die S =
Toss a coin S =
Toss two coins S =
Toss three coins S =
Roll two dice S = (1-1, 1-2, 1-3, 1-4, 1-5, 1-6,
_______________________________, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, _______________________________, 5-1, 5-2, 5-3, 5-4, 5-5, 5-6,
____________________)
Notation
________ denotes a probability
________ denote specific events
________ denotes the probability of event A occurring
Probability Limits The probability of an impossible event is _________. (My dog woof woof will start to fly)
The probability of an event that is certain to occur is ______. (Someone in this class is not a fan of math)
For any event A , the probability of A is between 0 and 1 inclusive. That is, 0 ≤ P ( A ) ≤ 1.
Possible Values for Probabilities
Basic Rules for Computing Probability Rule 1 : _________________________________________________________ Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P ( A ) is estimated as follows:
P(A) =
**Rule 2: __________________________________________ **(Requires ________________________)**** Assume that a given procedure has _____ different simple events and that each of those simple events has an ________________ chance of occurring. If event A can occur in s of these n ways, then
P(A) =
Rule 3: ________________________________
P ( A ), the probability of event A , is _____________________by using ____________________ of the relevant
___________________________.
Example 1 for Rule 1
Can you wiggle your ears? a) Since one can wiggle their ears or not, is the probability that one can wiggle their ears equal to 0.5?
b) Actually determine if you can you wiggle your ears? Then determine what the actual % of students in class than can wiggle their ears.
Example 2 for Rule 1 Do couples get engaged or not? If they are engaged, how long did they date before becoming engaged? A poll of 1000 couples conducted by Bruskin and Goldring Research gave the following information:
More than 2 350
1 to 2 years 210
Less than 1 year 240
Never engaged 200
Number of Couples
Length of Dating Time Before Engagement
Example 2 for Rule 2 Let event A = tossing a H when flipping a coin
P(A ) = ________________________
Let event A = tossing 2 heads (HH) when flipping 2 coins or getting 2 heads in a row when tossing 1 coin twice.
P(A ) = ________________________
Let event A = tossing 3 coins and getting 1 head and 2 tails.
P(A ) = ________________________
Example 3 for Rule 2 Bob is asked to construct a probability model for couples with three children. He determines that couples with three children can have all boys, all girls, or a mix of girls and boys. Because Bob considers three outcomes, he reasons the probability of each outcome is 1/3. What is wrong with Bob’s reasoning? Find the real probabilities of the three outcomes.
Example 4 for Rule 2 Two balls are drawn out of a bag without replacement containing two red balls and four green balls. a) List all simple events.
b) Determine the probability of drawing a red and a green ball.
Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability. Give example.
Definition The _______________________________________, consists of all outcomes in which the event A
Rounding Off Probabilities When expressing the value of a probability, either give the _____________________ or decimal or round off
final decimal results to ________________ significant digits. (Suggestion: When the probability is not a
simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.)
Let’s define some events: A: {randomly selecting an object that is square} P(A) = __________ B: {randomly selecting an object that is solid black} P(B) = __________ C: {randomly selecting an object that is an open circle} P(C) = __________
Find P(A U B) by the Addition rule:________________________________________________________________
Events A and B are disjoint or ___________________________ if they cannot occur at the same
time. (That is, disjoint events do not overlap.)
Complementary Events P ( A ) and P( A ) are disjoint
It is impossible for an event and its _____________________ to occur at the ________ time.
Titanic Mortality Men Women Boys Girls Total Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 Let event A = {Selecting a man} and event B = {Selecting a boy} P(A) = _______ P(B) = ________
Find P(A (^) U B)
Venn Diagram for Events That Are Not Disjoint
Venn Diagram for Disjoint Events
A: {randomly selecting an object that is square} P(A) = __________ B: {randomly selecting an object that is solid black} P(B) = __________ C: {randomly selecting an object that is an open circle} P(C) = __________
Find P(A (^) U C) = ______________________________
Since A ∩ C has no sample points we could modify the Addition rule: _______________________________
A: {randomly selecting an object that is square} B: {randomly selecting an object that is solid black} C: {randomly selecting an object that is an open circle}
Are events A and B mutually exclusive?______
Are events A and C mutually exclusive?______
Summary of the Addition Rule Additive Rule for events that are not mutually exclusive Additive Rule for events that are mutually exclusive
Key Point – Conditional Probability The probability for the second event B should take into account the fact that the first event A has already occurred.
Notation for Conditional Probability
_________________ represents the probability of event B occurring after it is assumed that event A has already
occurred (read B|A as “ B _____________ A .”)
Example 2 Two balls are drawn out of a bag without replacement containing two red balls and four green balls. a) List all the simple events.
b) Determine the probability of drawing a:
- red and a green ball
- 2 red balls
- 2 green balls
Independent Events Two events A and B are _____________________________________________________________
_______________________________________________________________________________________.
(Several events are similarly independent if the occurrence of any does not affect the probabilities of occurrence
of the others.) If A and B are not independent, they are said to be _____________________.
Formal Multiplication Rule P ( A and B ) = ___________________________________
Note that if A and B are ______________________________ events, _______ is really the same as ______.
Intuitive Multiplication Rule When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B , but be sure that the probability of event B takes into account the previous occurrence of event A.
Titanic
Mortality Men Women Boys Girls Total
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
- Find the probability of selecting one passenger who was a Man and someone who died. First define two events.
Are the events Mutually Exclusive? ____________
Are the events Independent? _______ Show why:
- Find the probability of selecting one passenger who was a Man and a boy. Use the two events defined above.
Mutually exclusive? ____________
Independent? _______ Show why
Applying the Multiplication Rule
