Intro to Probability, fundamental Principle of Counting, Permutation, Combination

1. Probability

2. Probability
• Experiment – an activity with observable
results.
• Outcomes – the results of an experiment.
• Sample Space – the set of all possible
different outcomes of an experiment.

3. Probability
Ex. a. Tossing a coin once
outcome = H S(n)=2
b. tossing three different coins together
outcome= HHH S(n)=8
c. rolling a die
outcome = 1 S(n) = 6
d. throwing a coin and a die together
outcome = H1 S(n)=12

4. Probability
Event – a subset of a sample space of an
experiment.
Subset – Let E be an event of the sample space
S. Since every outcome in E is an outcome in S,
we say that event E is a subset of S, denoted by
𝐸 ⊂ 𝑆.

5. Probability
The Union of two events
The union of two events E and F is event
𝐸 ∪ 𝐹
Venn Diagram

6. Probability
The Union of two events
Example if set A= { 1,2,3} and set B= { 3,4,5,6}
Then AUB=
{1,2,3,4,5,6}

7. Probability
The Intersection of two events
The intersection of two events E and F is the set
𝐸 ∩ 𝐹.

8. Probability
The Intersection of two events
Example if set A= { 1,2,3} and set B= { 3,4,5,6}
Then A∩ 𝐵 =
{3}
Example set C = { even integers} and set D={ odd
integers}
Then C ∩ 𝐷 =
Empty set or is called impossible event.
𝑜𝑟 ∅.

9. Probability
The complement of an event
The complement of an event E is the event
Ec.
Read as “E complement” is the set comprising
all the outcomes in the sample space S that are
not in E.

10. Probability
The complement of an event

11. Probability
The complement of an event
Example If set D = { number of dot/s in a die}
and set E = { 2,4,6 } then Ec is
Ec = { 1,3,5}

12. Probability
Mutually Exclusive Events
E and G are mutually exclusive
event
if 𝐸 ∩ 𝐺 = ∅

13. Probability
Mutually Exclusive Events

14. Probability
Example set E = { even numbers} and
set G={ odd numbers}
then E and G are mutually exclusive events.
or
C ∩ 𝑫 =

15. Probability
1) In rolling a pair of dice (one white and one black), we have the following
event.
A={(1,6)}
B={(1,6),(6,1)}
C={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)}
D={(1,1),(1,6),(6,1),(6,6)}
E={(1,1),(1,3),(5,5)}
Question
a) Which events are subsets of the other?
A ⊂ 𝐵, A ⊂C, A ⊂D, B ⊂D, A ⊂B ⊂D
b) What is AUB? B
c) What is A∩ 𝐵? A
d) What is 𝐵 ∩ 𝐷.B
e) Which events are mutually exclusive? A and E,B and E

16. Probability
2) In rolling a pair of dice, determine the events
M,N,P, and Q such that the sum of the numbers
are 4, 5,6, and 7, respectively.
M={(2,2),(1,3),(3,1)}
N={(2,3),(3,2),(1,4),(4,1)}
P={(3,3),(2,4),(4,2),(1,5),(5,1)}
Q={(3,4),(4,3),(2,5),(5,2),(1,6),(6,1)}

17. Probability – is the measure of how likely an
event is to appear.
P(E)=
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

18. Probability
1. The spinner may stop on any one of the eight
numbered sectors of the circles. Use the spinner
at the right to find each probability.
a. P(2)
b. P(white)
c. P(9)=
d. P(not white)=
e. P(white or Red)=

19. Probability
2. There are 3 red pens, 4 blue pens, 2 black
pens, and 5 green pens in a drawer. Suppose you
choose a pen at random.
a) What is the probability that the pen chosen is
red? 3/14
b) What is the probability that the pen chosen is
blue? 2/7
c) What is the probability that the pen chosen is
red or black? 5/14

20. Probability of Independent Events
• The outcome of an independent event is not
affected by the outcome of another.
If events A and B are independent, the probability
of both events occurring is found by multiplying the
probabilities of the events.
P(A and B) = P(A) x P(B)
or P(A∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)

21. Probability of Independent Events
Example 1. The spinner at the right is spun
twice. Find the probability of each outcome.
a) An A and B
b) a B and a C

22. Probability of Independent Events
Example 2. Every end of the month the faculty of Hayna
High School goes bowling at the Bowlingan sa Canto. On
one shelf of the bowling alley there are 6 green and 4 red
bowling balls. One teacher selects a bowling ball. A
second teacher then selects a ball from the same shelf.
What is the probability that each teacher picked a red
bowling ball if replacement is allowed?
Solution:
P(both red) = P(red) x P(red)
= (4/10)(4/10) = 4/25
The probability that both teachers selected a red ball is
4/25.

23. Probability of Dependent Events
Dependent Events
Two events are dependent if the outcome of
one of them has an effect on the outcome of
another.
The Probability of an event B occurring given that
an event A has already occurred is P(B/A), and read
as, “The probability of B given A”.
P(A and B) = P(A) x P(B/A)

24. Probability of dependent Events
Example 3.There are 5 blue marbles, 3 red
marbles, and 4 black marbles in a box.
a) What is P(blue)

25. Probability of dependent Events
Example 3.There are 5 blue marbles, 3 red
marbles, and 4 black marbles in a box.
b) What is P(red)

26. Probability of dependent Events
Example 3.There are 5 blue marbles, 3 red
marbles, and 4 black marbles in a box.
c) What is P(black)

27. Probability of dependent Events
Example 3.There are 5 blue marbles, 3 red
marbles, and 4 black marbles in a box.
d) What is P(blue and red) if there is no
replacement? 5/12 x3/11=5/44

28. Probability of dependent Events
Example 3.There are 5 blue marbles, 3 red
marbles, and 4 black marbles in a box.
e. ) What is P(blue and black) if there is no
replacement? 5/12 x 4/11 =5/33

29. Probability of dependent Events
Example 3.There are 5 blue marbles, 3 red
marbles, and 4 black marbles in a box.
f) What is P(blue and red) if there is
replacement? 5/12 x 3/12 =5/48

30. Probability of dependent Events
• There are eight white socks and five black
socks in a drawer.
a. What is the probability that you can pull out
a white socks?
b. If you pull one sock out of the drawer and
then another, what is the probability that you
can pull out 2 white socks?

31. Mutually Exclusive Events
M.E.E are events in which one or the other of
two events, but not both, can appear.
Rule: The Addition Rule
If A and B are M.E.E, then
P(A or B)=P(A) + P(B)

32. Mutually Exclusive Events
Ex. 1.What is the probability of drawing an ace
or a king from a standard deck of cards?
Solution:
P(ace or King) = P(ace) + P(king)
=
4
52
+
4
52
=
2
13

33. Mutually Exclusive Events
Example 2. There are 6 girls and 5 boys on the school paper
staff. A committee of 5 students is being selected at random
to design the editorial illustration of the pilot issue. What is
the probability that the committee will have at least 3 boys?
Solution:
P(at least 3 boys)=P(3 boys)+P(4 boys)+P(5 boys)
= 3b,2g 4b,1g 5b,0g
=
5𝐶3∙6𝐶2
11𝐶5
+
5𝐶4∙6𝐶1
11𝐶5
+
5𝐶5∙6𝐶0
11𝐶5
=
150
462
+
30
462
+
1
462
=
181
462
The probability that at 3 boys on the committee is
181
462

34. Probability of Inclusive Events
If two events, A and B, are inclusive, then the
probability that either A or B is the sum of their
probabilities decreased by the probability of
both appearing.
P(A or B) = P(A) + P(B)-P(A and B)

35. Probability of Inclusive Events
Example 1. One die is tossed. What is the
probability of tossing a 4 or a number greater
than 3?
Solution:
P(4 or > 3)=P(4)+P(>3)-P(4 and >3)
=
1
6
+
3
6
−
1
6
P(4 or > 3) =
1
2

36. Probability of Inclusive Events
Example 2. What is the probability of drawing a
King or a heart from a deck of cards?
Solution:
P(K or <3) = P(K) + P(<3) – P(King of hearts)
=
4
52
+
13
52
−
1
52
P(K or <3) =
4
13