Transcript Lesson 4 - West Virginia University

PROBABILITY EVERYDAY
Probability is an everyday occurrence in our lives.
 What is the probability it will rain today?
 What is the probability you will get a 90% or better on
your mid-term?
 What is the probability you will win the super lottery?
USE COMMON SENSE
When approaching a probability problem, it is
usually best to use your common sense. What do
you expect would happen in a given situation?
For example, if I give you a die and tell you to throw
it, what is the probability that you will roll a 5?
PROBABILITY FORMULA
The general formula for finding the probability of an
event is:
p(A) = Number of outcomes in the event
Total number of possible outcomes
PROBABILITY FACTS
The probability of an event is always between 0 and 1.00.
That is,
 If an event can never happen then p(A) = 0.
 If an event always happens then p(A) = 1.00
USEFUL TERMS
The following are some useful terms when doing probability
problems:
 Random sample
 Sampling with replacement
 Sampling without replacement
 Mutually exclusive
 Independent events
 Dependent events
RANDOM SAMPLE
In a random sample:
 Each individual in the population has an equal chance of
being selected.
 If more than one individual is to be selected for the
sample, there must be constant probability for each and
every selection.
SAMPLING
In sampling with replacement, an individual selected is
returned to the population before the next selection is
made.
In sampling without replacement, an individual selected is
not returned to the population before the next selection.
MUTUALLY EXCLUSIVE
Two events are mutually exclusive if they cannot occur
simultaneously. For example:
 A single roll of a die cannot result in a 2 AND a 5.
 A single card selected from a deck cannot be a Heart AND
a Diamond (mutually exclusive), but it can be a Heart AND
a Queen (not mutually exclusive).
INDEPENDENT EVENTS
Two events are independent if the outcome of one event
does not effect the probability of the second. For example:
 Rolling a single die twice (or rolling two dice
simultaneously) are independent events. What you get on
one roll does not effect the second roll.
Drawing two cards from a deck with replacement.
INDEPENDENT EVENTS
?
Conclusion—drawing two cards out of a deck without
replacement are NOT independent events.
DEPENDENT EVENTS
Two events are dependent if the outcome of one event
does effect the probability of the second. For example:
 Drawing two cards from a deck without replacement.
ADDITION RULES
General Addition Rule for finding p(A or B):
p(A or B) = p(A) + p(B) – p(A and B)
When A and B are mutually exclusive
p(A or B) = p(A) + p(B)
ADDITION RULES
General Addition Rule for finding p(A or B):
p(A or B) = p(A) + p(B) – p(A and B)
p(diamond) = 13/52
p(5) = 4/52
p(5 of diamonds) = 1/52
p(diamond OR 5) = 13/52 + 4/52 – 1/52 = 16/52
MULTIPLICATION RULES
General Multiplication Rule for finding p(A and B):
p(A and B) = p(A)p(B|A) where p(B|A) is the probability of
event B given that event A has already occurred.
When A and B are independent events:
p(A and B) = p(A)p(B)
When A and B are mutually exclusive, p(A and B) = 0
MULTIPLICATION RULES
p(A and B) = p(A)p(B|A) where p(B|A) is the probability of
event B given that event A has already occurred.
Event A is drawing a King
Event B is drawing a 5
p(A) = 4/52
p(B|A) = ¼
p(A and B) =
 4  1   4/  1  1
       
 52  4   52  /4  52
CONTINUOUS PROBABILITY
The formula to find the probability of an event from a
continuous normally distributed variable is:
X-m
z= s
PROBABILITY AND PROPORTION
The good news is that the probability of an event, that is a
single score or a set of scores, from a normal distribution is
exactly the same thing as the proportion. Use exactly the
same procedures, formulas, etc. that you used before. Just
refer to your answer as the probability of the event.