The document discusses counting techniques and probability. It introduces fundamental counting principles like the multiplication principle and tree diagrams to list and count outcomes of compound events. Examples include counting the possible tour schedules for a concert in 3 cities (6 outcomes) and food-sauce combinations at a stall selling 3 foods and 3 sauces (9 outcomes). It also explains how to use a tree diagram to count the 8 possible outcomes of tossing a coin 3 times.

1. COUNTING
TECHNIQUES AND
PROBABILITY

2. If the saying “ nothing is certain
except death and taxes” is true, then
all of the other events in our lives are
uncertain. For instance, will it rain
today? Will you pass the exam? Will
the package arrive on time? Will the
plane land safely? Will our team win
the game? All of these are uncertain
events and the study of uncertain
events is called probability. Finding
the probability of an event requires
counting the elements of the sample
space. Sometimes sample space is
so large that shortcuts are needed to
count all the possibilities. Hence, we
need to learn useful counting
techniques such as fundamental
principle of counting, permutation,
and combination
What is the chance
of getting struck by a
lightning, winning a
lotto, or getting a girl
you love say “YES”?

3. Tree diagrams and the
Fundamental Principle of
Counting
1.Sarah Geronimo is a popular concert artist.
Suppose she is planning a concert tour in
three cities – Manila, Cebu , and Davao. In
how many ways can she arrange her tour
schedule?If there is no restriction on the order of
performances, then Sarah may start in one of
the cities. After the first city is chosen, she may
choose from the two remaining cities as her
second stop. The remaining city is her last
stop.
The tree diagram below shows the possible
tour schedules,
First Stop. Second stop. Third
Stop
Manila Cebu. ----------
Davao
Davao ---------------
6 possible outcomes

4. 2.A FOOD STALL SELLS SQUID BALL, FISH
BALL, AND KIKIAM. THERE ARE ALSO THREE
CHOICES FOR THE SAUCE: SWEET, SPICY OR
SWEET AND SPICY. HOW MANY DIFFERENT
COMBINATIONS ARE POSSIBLE?

5. ■ 2. Food Choice Sauce Outcomes
■ squid ball sweet _________ SB , sweet
■ ( SB) spicy _________ SB , spicy
■ sweet n spicy _______SB , aweet n spicy
■ START Fish ball sweet __________FB , sweet
■ (FB) spicy __________FB , spicy
■ sweet n spicy_________FB , swet n spicy
■ kikiam sweet ________K , sweet
■ (K) spicy _______ K , spicy
■ Sweet n spicy _______ K , sweet n spicy
there are 9 possible outcomes

6. ■ 3. A coin is tossed thrice. Draw a tree diagram to illustrate the
possible outcomes.
■ First toss second toss third toss outcomes
■ H H HHH
■ H T T HHT
■ Start H ____ HTH
■ T _____ HTT
■ T H H _____THH
■ T _____THT
■ T H _____TTH
■ T _____TTT
there are 8 possible outcomes

7. WHEN ATASK CAN BE DONE INTWO OR MORE STAGES
AND EACH STAGE CAN BE DONE IN A NUMBER OF WAYS ,
TREE DIAGRAM HELP IN SHOWINGTHE POSSIBLE
CHOICES AND IN DETERMININGTHE NUMBER OF WAYS
THATTHEWHOLETASK CAN BE DONE.
,
WhenTHE COMBINATION OF ITEMS OR A SUCCESSION OF
EVENTS ARE CONSIDERED, EACH RESULT IS CALLED AN
OUTCOME. AN EVENT IS SUBSET OF ALL POSSIBLE
OUTCOME. A COMPOUND EVENT OCCURS WHEN AN EVENT
IS COMPOSED OFTWO OR MORE OUTCOMES, SUCH AS
FLIPPING A COIN FOLLOWED BY FLIPPING ANOTHER COIN.

8. GENERALIZATION:
■ THE FUNDAMENTAL PRINCIPLE OF COUNTING
■ In a compound event in which the first event may occur in n1 different ways, the
second event may occur in n2 different ways, and so on and the kth event may occur in
nk different ways, the total number of ways the compound event may occur is n1 * n2
* n3 ...... nk .
■ Ex 1. ConcertTour Schedule
■ First Stop Second Stop Third Stop Number of PossibleWays
■ 3 2 1 6
■ 2. Order in Food Stall
■ Food Choice Sauce Number of possibleWays
■ 3 3 9
■ 3.Tossing a CoinThrice
■ 1stToss 2ndToss 3rd toss number of PossibleWays
■ 2 2 2 8

9. PRACTICE EXERCISES:
■ TRYTHIS
■ 1. In a cafe , coffee can be ordered in three sizes, small (S) medium (M) , and
■ large (L), and in four flavors - Cappuccino (C) , Machiatto (M) Americano ( A) , and
■ Latte (L). In how many ways can you order coffee?
■ 2. Suppose an experiment consist of tossing a coin four times.
■ a. How many different outcomes are possible?
■ b. how many outcomes have at least 1 head?
■
■ c. How many outcomes have at most 2 heads?
■ d. How many different outcomes have exactly 3 heads
16
15, all outcomes,exceptTTTT have
at least one head
1 + 4 + 6 = 11
4
3 x 4 = 12
Hint:The outcomes with outmost 2 heads corresponds to the sequence with 0 , 1 or 2 heads

10. ASSIGNMENT:
■ “HAPPY BIRTHDAYTOYOU”
■ Andy asked his girlfriend to make all the decisions for their
date on her birthday. She will pick a restaurant and an activity for
the date.Andy will choose a gift for her.The local restaurants
include Korean , Chinese , and Italian.The activities she can
choose from are bowling, movies and putt-putt. Andy will buy
either chocolate or flowers.
■ 1. How many outcomes are there for these three decisions?
■ 2. Draw a tree diagram to illustrate the choices.