Post on 01-Nov-2014
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SECTION 7-1Basic Principles of Probability
Warm-upA drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.1. What is the probability that the sock is red?
2. What is the probability that the sock is not white?
3. What is the probability that the sock is green?
Warm-upA drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.1. What is the probability that the sock is red?
2. What is the probability that the sock is not white?
3. What is the probability that the sock is green?
rr + b + w
Warm-upA drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.1. What is the probability that the sock is red?
2. What is the probability that the sock is not white?
3. What is the probability that the sock is green?
rr + b + w
r + br + b + w
Warm-upA drawer contains r red socks, b blue socks, and w white socks.
Assume that you draw a sock at random from the drawer.1. What is the probability that the sock is red?
2. What is the probability that the sock is not white?
3. What is the probability that the sock is green?
rr + b + w
r + br + b + w
0
Probability theory:
Probability theory: The study of chance using math
Probability theory: The study of chance using math
Experiment:
Probability theory: The study of chance using math
Experiment: A situation with several possible results
Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome:
Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome: A possible result of an experiment
Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome: A possible result of an experiment
Sample Space:
Probability theory: The study of chance using math
Experiment: A situation with several possible results
Outcome: A possible result of an experiment
Sample Space: The set of the total number of possible outcomes of an experiment
Example 1A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the characteristics of the deck.
Example 1A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the characteristics of the deck.
The sample space is 52, since there are 52 cards.
Example 1A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the characteristics of the deck.
The sample space is 52, since there are 52 cards.
There are 26 red (13 hearts, 13 diamonds) and 26 black (13 clubs, 13 spades)
Example 1A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the characteristics of the deck.
The sample space is 52, since there are 52 cards.
There are 26 red (13 hearts, 13 diamonds) and 26 black (13 clubs, 13 spades)
Each suit has 4 face cards (J, Q, K, A) for a total of 16 in the deck
Example 1A standard deck of playing cards is used to draw a card at random.
a. Give the sample space and describe some of the characteristics of the deck.
The sample space is 52, since there are 52 cards.
There are 26 red (13 hearts, 13 diamonds) and 26 black (13 clubs, 13 spades)
Each suit has 4 face cards (J, Q, K, A) for a total of 16 in the deck
Anything else?
Example 1b. Identify the event “red face cards.”
Example 1b. Identify the event “red face cards.”
There are 8 red face cards:4 hearts
4 diamonds
Example 1b. Identify the event “red face cards.”
There are 8 red face cards:4 hearts
4 diamonds
{J♡, Q♡, K♡, A♡, J♢, Q♢, K♢, A♢}
Event:
Event: Any subset of the sample space or experiment
Event: Any subset of the sample space or experimentThis means that it is a particular occurrence of something happening
Event: Any subset of the sample space or experimentThis means that it is a particular occurrence of something happening
Probability that E occurs:
Event: Any subset of the sample space or experimentThis means that it is a particular occurrence of something happening
Probability that E occurs: The probability that the given event will occur
Event: Any subset of the sample space or experimentThis means that it is a particular occurrence of something happening
Probability that E occurs: The probability that the given event will occurSo, E is an event within our sample space S. To do this, we need to know the size of our sample space and the size of our favorable outcomes.
Event: Any subset of the sample space or experimentThis means that it is a particular occurrence of something happening
Probability that E occurs: The probability that the given event will occurSo, E is an event within our sample space S. To do this, we need to know the size of our sample space and the size of our favorable outcomes.
P(E) = # of favorable outcomes# of possible outcomes in S
Fair:
Fair: When each outcome is as likely as another
Fair: When each outcome is as likely as another
Biased:
Fair: When each outcome is as likely as another
Biased: When all outcomes are not equally likely to occur
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space:
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,6
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,6
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum:
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum: 7
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum: 7
P(7) =
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum: 7
P(7) = # of sum of 7# of possible sums
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum: 7
P(7) = # of sum of 7# of possible sums
=636
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum: 7
P(7) = # of sum of 7# of possible sums
=636
=16
Example 2Suppose two fair dice are rolled. What sum has the highest
probability of occurring? What is its probability?
Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6
Highest Sum: 7
P(7) = # of sum of 7# of possible sums
=636
=16
= 16 2/3% chance of rolling a 7
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12)
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12) =1236
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12) =1236
=13
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12) =1236
=13
P(2, 5, 8, or 11)
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12) =1236
=13
P(2, 5, 8, or 11) =1236
=13
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12) =1236
=13
P(2, 5, 8, or 11) =1236
=13
P(1, 4, 7, or 10) = 1236
=13
Example 3Persons A, B, and C were playing a game and needed to deicide who would go first. Person A suggested tossing two dice. If a 3,
6, 9, or 12 appeared, person A would go first. If a 2, 5, 8, or 11 appeared, person B would go first. If a 1, 4, 7, or 10 appeared,
person C would go first. Is this fair? Why or why not?
P(3, 6, 9, or 12) =1236
=13
P(2, 5, 8, or 11) =1236
=13
P(1, 4, 7, or 10) = 1236
=13
This is fair since all of the probabilities are the same
Theorem: Basic Properties of Probability
Theorem: Basic Properties of Probability
Let S be the sample space associated with an experiment and P(E) be the probability of E.
Theorem: Basic Properties of Probability
0 ≤ P(E) ≤ 1
Let S be the sample space associated with an experiment and P(E) be the probability of E.
Theorem: Basic Properties of Probability
0 ≤ P(E) ≤ 1
If E = S, then P(E) = 1
Let S be the sample space associated with an experiment and P(E) be the probability of E.
Theorem: Basic Properties of Probability
0 ≤ P(E) ≤ 1
If E = S, then P(E) = 1
If E = ∅, then P(E) = 0
Let S be the sample space associated with an experiment and P(E) be the probability of E.
Example 4If births of boys and girls are assumed equally likely, what is the
probability that a family with four children has all girls?
Example 4If births of boys and girls are assumed equally likely, what is the
probability that a family with four children has all girls?Make a list of all possible outcomes
BBBB BBBG BBGB BGBB
GBBB
BBGG BGGB GGBB BGBG
BGGB
GGGBGGBG
GBGG BGGG
GGGG
GBGB
Example 4If births of boys and girls are assumed equally likely, what is the
probability that a family with four children has all girls?Make a list of all possible outcomes
BBBB BBBG BBGB BGBB
GBBB
BBGG BGGB GGBB BGBG
BGGB
GGGBGGBG
GBGG BGGG
GGGG
GBGB
One situation out of the sixteen gives 4 girls, so 1/16, or 6.25%
Homework:
Homework:
p. 431 #1-23