Notes 7-1

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?





rr + b + w





rr + b + w

r + br + b + w





rr + b + w

r + br + b + w

0

Probability theory:

Probability theory: The study of chance using math


Experiment:


Experiment: A situation with several possible results



Outcome:



Outcome: A possible result of an experiment




Sample Space:




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.



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





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.”


There are 8 red face cards:4 hearts

4 diamonds


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


Probability that E occurs:


Probability that E occurs: The probability that the given event will occur


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.


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: When each outcome is as likely as another


Biased:


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?



Sample Space:



Sample Space: 1,1



Sample Space: 1,1 1,2



Sample Space: 1,1 1,2 1,3



Sample Space: 1,1 1,2 1,3 1,4



Sample Space: 1,1 1,2 1,3 1,4 1,5



Sample Space: 1,1 1,2 1,3 1,4 1,5 1,6



Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1



Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2



Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3



Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4



Sample Space: 1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5



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



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



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



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:



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



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) =



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



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


=636



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


=636

=16



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


=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?




P(3, 6, 9, or 12)




P(3, 6, 9, or 12) =1236




P(3, 6, 9, or 12) =1236

=13




P(3, 6, 9, or 12) =1236

=13

P(2, 5, 8, or 11)




P(3, 6, 9, or 12) =1236

=13

P(2, 5, 8, or 11) =1236

=13




P(3, 6, 9, or 12) =1236

=13

P(2, 5, 8, or 11) =1236

=13

P(1, 4, 7, or 10) = 1236

=13




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


Let S be the sample space associated with an experiment and P(E) be the probability of E.


0 ≤ P(E) ≤ 1



0 ≤ P(E) ≤ 1

If E = S, then P(E) = 1



0 ≤ P(E) ≤ 1

If E = S, then P(E) = 1

If E = ∅, then P(E) = 0


Example 4If births of boys and girls are assumed equally likely, what is the

probability that a family with four children has all girls?


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


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

Notes 7-1

Technology

Transcript of Notes 7-1