ST 351

Probability Rules – Classical Probability

Sections 4.2 and 4.5

 

Introduction

 

1.         Define sample space.

 

 

 

 

2.         Define event.

 

 

 

 

3.         Define equally likely events:

 

 

 

4.         Let A be an event.  Then

 

ex:  To figure the probability of getting a head on any toss of a fair coin, answer the following questions:

 

a)         What is the sample space?  How many outcomes are there in the sample space?

 

 

 

b)         What is the event of interest?  How many outcomes are in this event?

 

 

 

c)         Are the events “tossing a head” and “tossing a tail” equally likely?

 

 

 

d)         Determine P(tossing a head):

 

 

 

 

 


Probability Rules

 

1.         Probability of any event occurring [P(A) where A is an event] is always between 0 and 1, including 0 and 1.  That is, 0  P(A)  1

 

2.         All possible outcomes together must have probability 1.  P(S) = 1 where S is the sample space.

 

3.         P(Ac) = 1 - P(A).   Ac is the complement of the event A.  Complement means “not occurring” so Ac means event A is not occurring.

 

4.         If two events, call them A and B, are disjoint or mutually exclusive (meaning they have no outcomes in common), then P(A or B) = P(A) + P(B).  This is sometimes written as

            Note: since the two events are disjoint, the word “or” means “one event occurring or the other event occurring”.

 

5.         If two events, call them A and B, are independent of each other (meaning the outcome of event A does not depend in any way on the outcome of event B and vice versa), then

P(A and B) = P(A) x P(B).  (Note: P(A) x P(B) is often written simply as P(A)P(B) .)

This is sometimes written as

Note:  If P(A) P(B) = P(A and B), then A and B are independent.

Note 2: the word “and” means that both A and B occur at the same time!

 

6.         General addition rule for the union of two events: 

P(A or B) = P(A) + P(B) - P(A and B). 

Note 1:  This is the case when A and B are not disjoint.  Here, “A or B” means event A occurs, event B occurs, or both A and B occur at the same time!

Note 2:  This is sometimes written as

 

7.         Conditional Probability.  P(B|A) = P(A and B) 

                                                                  P(A)

P(B|A) is read, “the probability of event B given event A has already occurred.”

Note:  P(A|B) = P(A and B)

                                         P(B)

 

 

 

Examples

 

1.         Question 4.13:

All human blood can be “ABO-typed” as one of O, A, B, or AB, but the distribution of the types varies a bit among groups of people.  Here is the distribution of the blood types for a randomly chosen person in the United States:

                       

Blood type

O

A

B

AB

Probability

0.45

0.40

0.11

?

 

a)         (Rules 1 and 2) What is the probability of type AB blood in the United States?

 

 

 

 

 

b)         (Rule 4) Maria has type B blood.  She can safely receive blood transfusions from people with blood types O and B.  What is the probability that a randomly chosen American can donate blood to Maria?

 

 

 

 

 

 

 

 

 

2.         Question 4.18:

            Choose an acre of land in Canada at random.  The probability is 0.35 that it is forest and 0.03 that it is pasture.  (Assume it can’t be both.)

 

            a)         Draw a Venn diagram that represents these events.

 

Note: A Venn diagram can be a useful way of representing events and can be helpful in determining probabilities.  The Venn diagram is a rectangular box.  The area inside the box represents the sample space.  Circles are placed inside the box.  Each circle represents an event.  If two events are disjoint, the two circles representing those two events will NOT overlap.  However, if the two events have outcomes in common (not disjoint), then the two circles will overlap with the overlapping part representing the event “A and B”. 

 

-  What are the two events in this problem?  Are they disjoint?

 

 

 

 

 

                       

 

 

 

 

 

 

 

b)         (Rule 3) What is the probability that the acre chosen is not forested?  In your Venn diagram above, draw vertical lines to represent the event “not forested”.

 

 

 

 

 

 

c)         (Rule 4) What is the probability that the acre chosen is either forest or pasture?  In your  Venn diagram above, shade lightly the event “forest or pasture”.

 

 

 

 

 

d)         What is the probability that a randomly chosen acre in Canada is something other than forest or pasture?  What rule(s) are you using here?  Draw vertical lines in your Venn diagram to represent “something other than forest or pasture”.

 

 

 

 

 

 

e)         What is the sum of the probabilities of the events “forested”, “pasture”, “something other than forest or pasture”?  Can there be any other events beside these?

 

 

 

 

 

3.         (Rule 5) What’s the probability of getting heads on two tosses of a coin?

 

 

 

 

 

 

4.         (Rule 5) Shaquille O’Neal’s career free throw percentage is 53.3%.  What is the probability of Shaq making two consecutive free throws (assuming each free throw attempt is independent, which can be legitimately argued)?        

 

 

 

 

 

 

 

 

5.         A deck of playing cards consists of 52 cards divided evenly into 4 suits (hearts, diamonds, spades, and clubs).  Each suit contains an ace, two, three, …, ten, jack, queen, and king.

 

            a)         If one card is drawn from the full deck, what is the probability it is a club?

 

 

            b)         If one card is drawn from the full deck, what is the probability it is an ace?

 

 

c)         If one card is drawn from the full deck, what is the probability it is the ace of clubs (that is, an ace AND a club)?  (What rule can you use to find this probability?)

 

 

 

d)         If one card is drawn from the full deck, what is the probability it is an ace OR a club?  (Note that the word OR here means an ace, a club, or both an ace and a club.)  What rule are you using here?

 

 

 

 

 

 

6.         Question 4.86:

Call a household prosperous if its income exceeds $100,000.  Call the household educated if the householder completed college.  Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated.  According to the Current Population Survey, P(A) = 0.134, P(B) = 0.254, and the probability that household is both prosperous and educated is P(A and B) = 0.080.  Find P(A or B), the probability that the household selected is either prosperous or educated (or both)?

 

 

 

 

 

 

 

 

 

7.         Question 4.94 

The table below gives the counts (in thousands) of earned degrees in the United States in the 2001-2002 academic year, classified by education level and sex of the degree recipient.  Use these data to answer the following questions.

           

 

Bachelors

Master’s

Professional

Doctorate

Total

Female

645

227

32

18

  922

Male

505

161

40

26

  732

Total

1150

388

72

44

1654

 

            a.         What is the probability that a randomly chosen degree recipient is a man? A woman?

 

 

 

 

 

 

 

 

b.         What is the probability that a randomly chosen degree recipient is a female with a professional degree (that is, is a female AND has a professional degree)?

 

 

 

 

 

 

c.         Given that a randomly chosen person is a woman, what is the probability that she has a professional degree?

 

 

 

 

 

 

 

            d.         What proportion of professional degree recipients are women?