Download Chapter 6: Probability

Chapter 6: Probability
Adam Turry
Jordan Furr
Ansley Orgeron
Mackenzie Kruse
What is a simulation?
• A simulation is the
imitation of chance
behavior, based on a
model that accurately
reflects the
phenomenon under
consideration.
• The calculator is a
great way to carry out
a simulation by using
random integer
Steps of Simulations
•
•
•
•
•
State the problem
State the assumptions
Assign digits to represent outcomes
Simulate multiple repetions
Say your conclusion
Probability Models
• A random phenomenon has outcomes that we cannot
predict but have a regular distribution with many
repetitions.
• The probability of an event is the proportion of times the
event occurs in many repeated trials of a random
phenomenon.
• A probability model for a random phenomenon consists of
a sample space S and an assignment of probabilities.
• The sample space S is the set of all possible outcomes of
the random phenomenon.
• Events are sets of outcomes.
• A number P(A) is assigned to an event A as its probability.
Vocabulary
• A tree diagram is a simple way to look at the
possible outcomes of a probability sample.
Each of the outcomes is a branch on the
“tree.”
• The multiplication principle says is you can do
one task in n1 number of ways and another
task in n2 number of ways, then both tasks
can be done in n1 x n2 number of ways.
More Vocabulary
• Sampling with replacement takes place when you select an object
and replace it before the next selection.
– Replacing a marble into the bag before picking another
• Sampling without replacement, the probabilities change for each
new selection.
– Taking a marble out of the bag and leaving it out before selecting
another.
• The complement Ac of an event A consists of the outcomes that are
not in A.
• Events are disjoint (mutually exclusive) if they have no outcomes in
common.
• Events are independent if knowing that one event occurs does not
change the probability we would assign the other event.
• A Venn diagram shows events as disjoint of intersecting regions.
Rules of Probability
1. Legitimate Values- 0≤P(A) ≤1 for any event A
2. Total Probability- P(S) =1 for the sample space S.
3. Addition rule: if events a and B are disjoint, then P(A
or B) = P(A U B) = P(A) + P(B).
4. Complement rule: For any event A, P(Ac) = 1- P(A).
5. Multiplication rule : if events A and B are
independent, then P(A and B) = P (A∩B) = P(A)P(B).
6. General Addition Rule- (A U B) = P(A)+P(B)- (A∩B)
General Probability Rules
• Intersection- (A∩B)
contains all the outcomes
that are in set A, and B
• Union-(A U B) contains all
outcome of A or B, or in
both A and B.
• Conditional Probability- the
event of P(B) occurring
given P(A) has occurred.
Formula= P(A∩B)/P(A)
Venn Diagram
• Venn diagrams are word problems that
generally help you find probabilities of the
union and/or intersection of two events.
• Example:
Helpful Hints
• Disjoint(mutually exclusive)- if
A and B are disjoint, the
P(A∩B)=0
– The general addition rule for
unions is then the special
addition rule P(A U B)= P(A) +
P(B)
• A and B are independent
when P(B│A)=P(B)
– Multiplication rule for
intersection then becomes
P(A∩B)=P(A) P(B)