Transcript Introduction to Probability

MATH 1107
Elementary Statistics
Lecture 4
Introduction to Probability
Introduction to Probability
• Who is George Bush’s favorite friend?
– Monica
– Phoebe
– Rachel
– Joey
– Chandler
– Ross
• How many times has Dr. Priestley jumped out of an airplane?
– 0
– 1
– 2
– 3 or more
Introduction to Probability
Unless you have insight into these questions, you
would have to guess randomly. Therefore, the
probability that you are correct is 1/6 or 16.67% for
the first question and ¼ or 25% for the second
question.
Introduction to Probability
Terms used in analysis of probabilities:
• experiment
• event (represented by a capital letter – p(A))
• sample space
Introduction to Probability
Using a deck of cards as an example:
• Pulling a card from the deck is an experiment
• The card selected represents the event
• Each individual card together represents the sample space.
Describe the probability of selecting a heart
 Sample Space = {H,S,D,C}
 P(H) = ¼
Describe the probability of selecting the Jack of Clubs
 Sample Space = {2H, 3H, 4H…JC, QC, KC}
 P(JC) = 1/52
Introduction to Probability
Events in a Sample Space must be mutually exclusive and
collectively exhaustive.
Using a fair die, develop the sample space ensuring that all
events are mutually exclusive and collectively exhaustive.
What are two events in a die experiment that are mutually
exclusive? What are their individual and collective probabilities?
Introduction to Probability
In Class exercise: An experiment consists of flipping a fair coin
three times. Compute the probabilities of the following
events:
(A) Exactly one head will appear
(B) At least one head will appear
(C) Tails never comes up
Introduction to Probability
Step 1: Compute the mutually exclusive, collectively exhaustive
sample space:
HHH
THH
HTH
THT
HTT
TTH
HHT
TTT
Step 2: Determine how many ways each event can occur:
A: HTT, THT, TTH
B: HHH, HTH, HTT, HHT, THH, THT, TTH
C: HHH
Step 3: Divide the number of possibilities for each event by the
total number in the sample space:
A: 3/8 = 37.5
B: 7/8 = 87.5
C: 1/8 = 12.5
Introduction to Probability
In Class exercise: A pair of fair dice are rolled. Compute the
following events:
(A)The sum of the two dice is 7
(B)The sum of the two dice is 5
(C) The sum of the two dice is even
Introduction to Probability
Step 1: Compute the mutually exclusive, collectively exhaustive
sample space:
{(1,1), (1,2), (1,3)…(6,4), (6,5), (6,6)}
Step 2: Determine how many ways each event can occur:
A: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
B: (1,4), (2,3), (3,2), (4,1)
C: (1,1), (1,3), (1,5)…(6,2), (6,4), (6,6)
Step 3: Divide the number of possibilities for each event by the
total number in the sample space:
A: 6/36 = 16.67%
B: 4/36 = 11.11%
C: 18/36 = 50%
Introduction to Probability
Another way to discuss probabilities is in terms of “odds”. Odds
are stated as the ratio of the probability of an event occurring and
the probability of an event not occurring.
The odds of P(A) occurring is stated as:
P(A)/P(B) or b:a. Note that the event in question occurring is
always stated second.
Odds are generally used in gambling, lotteries, etc because it is a
convenient way to describe the payoffs.
Introduction to Probability
For example, the commonly accepted odds for the top five teams
to win the 2003 NCAA tournament were:
Duke 4:1
Maryland 5:1
Kansas 5:1
Cincinnati 7:1
Oklahoma 8:1