Download Basic Prob Rules

Probability and event will occur
The relative frequency with which an event can be expected to occur.
3 Ways to Obtain the Probability of an Event
1. Experimentally / Empirically  P’(A)
2. Theoretically  P(A)
3. Subjectively  Opinion
Empirical (Observed) Probability
Example: If you roll a die 40 times and 9 of the
rolls result in a “5”. What empirical probability
is observed for the event “5”?
Answer:
P' ( A) 
Number of times A occured
Number of trials
n( A)
P' ( A) 
n
probabilit y of A 
9
40
Sample Space
The set of all possible outcomes of an experiment, denoted by S.
Represented in many forms:
1 – List
2 – Tree diagram
3 – Lattice grid
Others…
The individual outcomes of a sample space are called sample points.
n(S) is the number of sample points in S.
Event
Any subset of the sample space. If A is an event, then n(A) is the number of sample points that belong to
event A.
Example: A single coin is tossed once and the outcome (H or T) is recorded.
What is the Sample Space, S?
S={H,T}
What is the number of sample points in S, n(S)?
n(S) = 2
Example: Two coins, one penny and one nickel, are tossed simultaneously, and the outcome is recorded in
ordered pair notation, (penny, nickel).
Nickel
What is the Sample Space, S?
H
Penny
List: S = {HH, HT, TH, TT}
Tree Diagram:
H
T
H
T
What is the number of sample points in S, n(S)?
T
n(S)=4
Example: A box contains three balls (one red, one blue, one white). Two balls are drawn from the box with
replacement and the color is recorded.
2nd Draw
What is the Sample Space, S?
List:
Tree Diagram:
S = {RR, RB, RW, BR, BB,
R
1st Draw
R
W
R
BW, WR, WB, WW}
B
What is the number of sample points in S, n(S)?
n(S) = 9
B
B
W
R
W
B
W
Theoretical Probability
In a sample space containing sample points that are equally likely to occur, the probability P(A), of an event A
is the ratio of the number n(A) of points that satisfy the definition of A to the number n(S) of sample points in
the entire sample space.
P( A) 
number of times A occurs in sample space
number of elements in sample space
P( A) 
n( A)
n( S )
Properties of Probability
Property 1
0  P( A)  1
* Probability of 0 means the event cannot occur.
*Probability of 1 means the event will always occur.
Property 2
P( S )  1
OR
 Probabilit ies  1
*If you add all the probabilities of each sample point in the space, then the total probability must equal 1.
Property 3
P( A)  1  P( Ac )
P( Ac )  1  P( A)
*Useful when being asked the probability of “at least one” or “not”.
Mutually Exclusive Events
Events defined in such a way that the occurrence of one event precludes the occurrence of any of the other
events.
*Two events are mutually exclusive if they cannot occur at the same time.
Example:
Event A: It rained on Tuesday.
Event B: It did not rain on Tuesday.
Venn Diagram
A
Rain
Example:
Consider your senior class.
Event A: Male
Event B: Female
Event C: Going to college
B
No
Rain
Venn Diagram
A
Male
A&C
Males going
to college
Which events are mutually exclusive? A&B
Which are not? A&C, B&C
B
Female
C
College
B&C
Females
going to
college
Addition Rule for Events that are Mutually Exclusive
Let A and B be two events defined in a sample space, S. If A and B are mutually exclusive events, then
P(A or B) = P(A) + P(B)
Example: A group of students from Georgetown and Horry Counties are chosen to participate in the national
geography bee.
10 students from Waccamaw, 7 students from St. James, 3 students from Low Country, 20 students from
Georgetown High
One student will win the bee.
Find the probability the winner is from Waccamaw or Georgetown.
P(W U G) = P(W) + P(G)

10 20

 0.75
40 40
Addition Rule for Events that are NOT Mutually Exclusive
Let A and B be two events defined in a sample space, S. Then
P(A or B) = P(A) + P(B) – P(A and B)
Example: A group of 200 college students consists of 140 full-time students (80 female and 60 male) and 60
part-time students (40 female and 20 male). One student is selected at random.
Find the probability that the student selected is full-time or female.
P(FT U F) = P(FT) + P(F) – P(both)

140 120 80


 0.9
200 200 200
Find the probability that the student selected is full-time or male.
P(FT U M) = P(FT) + P(M) – P(both)

140 80
60


 0.8
200 200 200
Independent Events
Two events A and B are independent if and only if the occurrence (or nonoccurrence) of one does not affect
the probability of the other.
Example: “Unrelated trials”
-Flip a coin AND draw a card.
-Choose a ball from a box, replace it, and choose a second.
-Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
-Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.
-Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.
Multiplication Rule for Independent Events
Let A and B be independent events in S.
P(A and B) = P(A)P(B)
Example: Two cards are drawn from a standard deck with replacement. What is the probability a
King and a Queen are chosen?
4
4
2


 0.1538
P(K and Q) =
52 52 13
Dependent Events
Two events A and B are dependent if the occurrence (or nonoccurrence) of one affects the probability of the
other.
Example:
Choose a card from a deck, do not replace it, AND choose a second card.
Roll dice- a sum of 10 results AND dice show a double
A box contains 3 white marbles and 4 black marbles. What is the probability of drawing 2 black marbles and 1
white marble in succession without replacement?
Slips numbered 1 through 9 are placed in a box. 2 slips are drawn, without replacement.
Conditional Probability (necessary for dependent events)
P (A l B) represents the probability that A will occur given that B has occurred. This is called the conditional
probability.
Formula:
P( A B) 
P( A  B)
P( B)
in words
Probability of A given B =
probabilit y of both A and B
probabilit y of B
P( A B) 
n( AandB)
n( B )
in words
Probability of A given B =
number of times A and B both occur
number of times B occurs
Example: P(A) = 0.2, P(B) = 0.6, P(A and B) = 0.2
P( A B) 
P( A  B) 0.2

 0.12
P( B)
0.6
Example: Conditional Probability – A single die is rolled. S = {1, 2, 3, 4, 5, 6}
A is the event “a 4 has occurred”
B is the event “an even number has occurred”
Find the probability of A given B.
P( A B) 
number of times A and B both occur 1

number of times B occurs
3
OR
Independent Events
Two events A and B are independent events if and only if
P(A l B) = P(A)
OR
P(B l A) = P(B)
Example: P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. Show that A and B are independent.
P(A l B) =
=
P ( A and B)
P(B)
0.2
 0 .5
0.4
P(A) = 0.5
P(A l B) = P(A)
Independent
General Multiplication Rule
Let A and B be events defined in a sample space S. Then
P ( A  B)  P(A)P(B A)
Example: Two cards are drawn without replacement. What is the probability a King and Queen are chosen?
P ( K  Q)  P(K)P(Q K)
4 4
 
52  51 
 0.0060

EXTRA PRACTICE
True/False
1. The probabilities of complementary events always add up to 1.
2. If A and B are mutually exclusive events, then if A has occurred, B may also occur.
3. If P(A)=0.3, P(B)=0.6, and P(A and B)=0.18, then A and B are independent.
4. If P(A)=0.4, P(B)=0.3, and P(A and B)=0.15, then P(BlA)=0.45
Multiple Choice
5. Suppose A and B are independent events with P(A) = 0.30 and P(B) = 0.50, then P(A and B) is…
a. 0.80
b. 0.60
c. 0.20
d. 0.15
6. A coin is tossed two times. What is the probability of getting tails both times?
a. 1/4
b. 1/3
c. 1/2
d. ¾
7. There are 10 cards in a deck. The cards are numbered from 1 through 10. If you draw one card, do not
replace it, and then draw a second card, what is the probability of both cards being even numbered?
a. 1/4
b. 2/9
c. 1/5
d. 19/90
8. You draw a card from a standard deck of 52 cards, keep it, and draw a second card. What is the probability
of both cards being clubs?
a. 1/4
b. 1/16
c. 1/17
d. 99/202
9. What value would be needed to complete the following probability distribution?
A
P(A)
a. 1/5
b. 1/12
0
1/3
1
1/8
2
1/8
3
?
4
1/6
c. 1/4
d. 1/8
Short Answer
10. In a survey, 100 students were asked “Do you prefer to watch television or play a sport?” Of the 46
boys in the survey, 33 said they would choose sport, while 29 girls said sport.
Boys
Girls
Sport
33
29
Total
46
Total
Television
100
Find the probability that…
a. a student selected at random prefers to watch television.
b. a student prefers to watch television, given that the student is a boy.
______
11. If one card is drawn from a standard deck of 52 cards, find the probability of getting these results, with
replacement.
a. An ace:
b. An 8 or a 10:
c. A 3 or a club:
d. A black king:
e. A black card or an ace:
12. A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the
other, without replacement. The first two apples are green. What is the probability that the third apple is red?
13. The probability that an individual will contract a particular disease is 0.005. Past experience
reveals that the probability that an individual who contracts the disease will make a complete recovery
is 0.68. Find the probability that a randomly selected individual contracts the disease and does not
make a complete recovery.
14. A box contains five red, three blue, and two white poker chips. Two are selected without
replacement.
a. Draw a tree diagram (with branches showing appropriate probabilities) to represent
this situation.
b. Find the probability that both poker chips are the same color.
15. Three cards are randomly selected from a standard deck of 52 cards. Find the probability that all
three cards are red, if they are selected…
a. with replacement.
b. without replacement.
c.
16. A box contains 3 defective units and 17 non-defective units. Two units are selected from the box
without replacement.
a. What is the probability that the second unit is defective given that the first one selected
was defective?
b. What is the probability that neither is defective?