Transcript Fibonacci*s Numbers

PROBABILITY
THE PROBABILITY OF AN EVENT E
EX1)

Two fair dice are rolled. What is the probability
that the sum of the numbers on the dice is 10?
EX2)
(1) For a fair die, what is the probability of an odd
number?
(2)Suppose that a die is loaded so that the numbers
2 through 6 are equally likely to appear, but that
1 is three times as likely as any other number to
appear. What is the probability of an odd
number?
THE PROBABILITY OF THE COMPLEMENT
EVENT
Let E be an event. The probability of E', the
complement of E, satisfies
P( E )  P( E ')  1
EX3) BIRTHDAY PROBLEM

Find the probability that among n persons, at
least two people have same birthdays. Assume
that all months and dates are equally likely, and
ignore Feb. 29 birthdays.
THE PROBABILITY OF THE UNION

The probability of the event A and B :
P( A B)

The probability of the event A or B :
P( A B)
P( A B)  P( A)  P( B)  P( A B)
EX4)
Two fair dice are rolled.
(1) What is the probability that a sum of 7 or 11
turns up?
(2) What is the probability of getting doubles (two
dice showing the same number) or a sum of 6?
CONDITIONAL PROBABILITY
The conditional probability of A given by B is
P( A | B) 
P( A B)
P( B)
P( B)  0
EX5)

A pointer is spun once on a circular spinner. The
probability assigned to the pointer landing on a
given integer is the ratio of the area of the
corresponding circular sector to the area of the
whole circle, as given in the table:
x
1
2
3
4
5
6
P(x )
.1
.2
.1
.1
.3
.2
(1) What is the probability of the pointer landing on
a prime number?
(2) What is the probability of the pointer landing on
a prime number, given that it landed on an odd
number?
EX6)

Suppose that city records produced the following
probability data on a driver being in an accident on the
last day of a Memorial Day weekend:
Rain
No Rain
R’
(1)
(2)
(3)
(4)
R
Accident A
No Accident
A’
Totals
.025
.335
.360
.015
.625
.640
Totals
.040
Find
the probability
of an.960
accident, rain1.000
or no rain.
Find the probability of rain, accident or no accident.
Find the probability of an accident and rain.
Find the probability of an accident, given rain
PRODUCT RULE
P( A B)  P( A) P( B | A)  P( B) P( A | B)


A and B are independent events if
P( A B)  P( A) P( B)
.
If A and
P(B
A |are
B) independent,
P( A)
P( B | Athen
)  P( B)
EX7) TESTING FOR INDEPENDENCE

In two tosses of a single fair coin, show that the
events “A head on the first toss” and “A head on
the second toss” are independent.
EX8) TESTING FOR INDEPENDENCE

A single card is drawn from a standard 52-card
deck. Test the following events for independence.
A=the drawn card is a spade
B=the drawn card is a face card
A=the drawn card is a spade
B=the drawn card is a face card