Probability

PROBABILITY
Prepared by:
LITILLE MAE D. BALAGOSA
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Multiplicative Rules PROBABILITY
Conditional Probability
Baye’s Principle
Multiplication Rule 1
Two events A and B are independent events if the fact that A occurs does
not affect the probability of B occurring.
When two events are independent, the probability

Multiplicative
of both occurring is: Rules
P(A and B) = P(A) P(B)
Examples
1. A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.
P(head and 4) = P(head) P(4) =
2. A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting
a queen and then an ace.
P(queen and ace) = P(queen) P(ace) =
3. An urn contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and its color noted. Then it
is replaced. A second ball is selected and its color noted. Find the probability of each of these.
a. Selecting 2 blue balls
b. Selecting 1 blue ball and then 1 white ball
c. Selecting 1 red ball and then 1 blue ball
a. P(blue and blue) = P(blue) P(blue) =

b. P(blue and white) = P(blue) P(white) =
c. P(red and blue) = P(red) P(blue) =
Two events A and B are independent events if the fact that A occurs does
not affect the probability of B occurring.
When two events are independent, the probability

of both occurring is:
P(A and B) = P(A) P(B) Multiplicative

NOTE:
Rules
Multiplication rule 1 can be extended to three or more independent events
by using the formula
P(A and B and C and . . . and K) P(A) P(B) P(C) . . . P(K)
When a small sample is selected from a large population and the subjects are not
replaced, the probability of the event occurring changes so slightly that for the
most part, it is considered to remain the same.
Examples
1. A Harris poll found that 46% of Americans say they suffer great stress at least once a week. If three
people are selected at random, find the probability that all three will say that they suffer great stress at
least once a week..
Let S denote stress
P(S and S and S) = P(S) P(S) P(S) =
2. Approximately 9% of men have a type of color blindness that prevents them from distinguishing
between red and green. If 3 men are selected at random, find the probability that all of them will have this
type of red-green color blindness.
Let C denote red-green color blindness
P(C and C and C) = P(C) P(C) P(C) =

When the outcome or occurrence of the first event affects the outcome or
occurrence of the second event in such a way that the probability is changed, the
events are said to be dependent events.
Multiplicative
When two events are dependent, the probability of Rules
both occurring is:
P(A and B) = P(A) P(B/A)
Examples
1. A bag contains 3 red balls and 3 black balls. One ball is drawn from the bag, set aside, and then a
second ball is drawn. Find the probability of getting 2 red balls.
Let
A = the first red ball
B = the second red ball
P(A and B) = P(A) P(B/A)
2. World Wide Insurance Company found that 53% of the residents of a city had homeowner’s insurance
(H) with the company. Of these clients, 27% also had automobile insurance (A) with the company. If a
resident is selected at random, find the probability that the resident has both homeowner’s and automobile
insurance with World Wide Insurance Company.
P(H and A) = P(H) P(A/H) =

Examples
3. Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events.
a. Getting 3 jacks
b. Getting an ace, a king, and a queen in order
c. Getting a club, a spade, and a heart in order
d. Getting 3 clubs
a. P(3 jacks) =
b. P(ace and king and queen) =
c. P(club and spade and heart) =
d. P(3 clubs) =
The conditional probability of an event B in relationship to an event A was
defined as the probability that event B occurs after event A has already occurred.
The probability that the second event B occurs given that the
first event A has occurred can be found by dividing the Conditional
probability that both events occurred by the probability that
the first event has occurred. The formula is: Probability
P(B/A) =
Examples
1. A box contains black chips and white chips. A person selects two chips without replacement. If the
probability of selecting a black chip and a white chip is , and the probability of selecting a black chip on
the first draw is , find the probability of selecting the white chip on the second draw, given that the first
chip selected was a black chip.
Let
B = selecting a black chip
W = selecting a white chip
P(W/B) =
Examples
2. The probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability
that Sam cannot find a legal parking space and has to park in the no parking zone is 0.20. On Tuesday, Sam
arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket.
Let
N = parking in a no parking zone
T = getting a ticket
P(T/N) =
For two events A and B, where event B follows event A, event A can occur
in , , . . . , mutually exclusive ways, and event B can occur in , , . . . ,
mutually exclusive ways,
Baye’s
for any specific events and Principle
Examples
1. On a game show, a contestant can select one of four boxes. Box 1 contains one P100 bill and nine P50
bills. Box 2 contains two P100 bills and eight P50 bills. Box 3 contains three P100 bills and seven P50
bills. Box 4 contains five P100 bills and five P50 bills. The contestant selects a box at random and selects a
bill from the box at random. If a P100 bill is selected, find the probability that it came from box 4.
Let
represent the boxes
100 and 50 represent the peso bills
Examples
2. A survey conducted by the Bureau of Labor Statistics found that approximately 27% of the high school
graduating class of 2010 went on a 2-year college, 41% went on to a 4-year college, and the remaining 32%
did not go on to college. Of those who went on to a 2-year college, 52% worked at the same time, 32% of
those going on to a 4-year college worked, and 78% of those who did not go on to college worked. What is
the percentage of those working that had not gone on to college?
Let
represent the graduates (went on to a 2-year college, 4-year college )and a
graduate that did not go to college
A represents the graduates that went on to work

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PROBABILITY

Multiplicative Rules PROBABILITY

When two events are independent, the probability

P(head and 4) = P(head) P(4) =

P(queen and ace) = P(queen) P(ace) =

a. P(blue and blue) = P(blue) P(blue) =

When two events are independent, the probability

P(A and B) = P(A) P(B) Multiplicative

Let S denote stress

P(S and S and S) = P(S) P(S) P(S) =

Let C denote red-green color blindness

P(C and C and C) = P(C) P(C) P(C) =

P(A and B) = P(A) P(B/A)

P(H and A) = P(H) P(A/H) =

b. P(ace and king and queen) =

c. P(club and spade and heart) =

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