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

( ) ( ') 1P E P E

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 :

The probability of the event A or B : ( )P A 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

( )( | ) ( ) 0

( )

P A BP A B P B

P B

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:

(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?

x 1 2 3 4 5 6

P(x ) .1 .2 .1 .1 .3 .2

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:

(1) Find the probability of an accident, rain or no rain.

(2) Find the probability of rain, accident or no accident.

(3) Find the probability of an accident and rain.

(4) Find the probability of an accident, given rain

Accident A

No Accident A’

Totals

Rain R

.025 .335 .360

No Rain R’

.015 .625 .640

Totals .040 .960 1.000

PRODUCT RULE

A and B are independent events if .

If A and B are independent, then

( ) ( ) ( | ) ( ) ( | )P A B P A P B A P B P A B

( ) ( ) ( )P A B P A P B

( | ) ( ) ( | ) ( )P A B P A P B A 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