course outline for probability

1. PROBABILITY
PREPARED BY CARMINA P. TILAD
UNIT IV: Problem Solving

2. OBJECTIVES:
• Probability
⚬ Counting Principle
⚬ Permutations and
Combinations
⚬ Binomial
Probability

3. COUNTING PRINCIPLE
If one event can occur in m ways, a second event can occur
in n ways, and a third event can occur in p ways, and so on,
then the sequence of events can occur in m x n x p x …
ways. This is known as the product rule or rule of product.
Example: How many possible outcomes are there if a die is
rolled once and a coin is tossed once?
6 x 2 = 12.
There are 12 possible outcomes when a die is rolled
once, and a coin is tossed once.

4. COUNTING PRINCIPLE
Example: How many possible outcomes are there if a die is
rolled once and a coin is tossed once?
There are 12 possible outcomes when a die is rolled
once, and a coin is tossed once.
Die roll/Toss coin Heads Tails
1 (1, heads) (1, Tails)
2 (2, heads) (2, Tails)
3 (3, heads) (3, Tails)
4 (4, heads) (4, Tails)
5 (5, heads) (5, Tails)
6 (6, heads) (6, Tails)
Using Table

5. MORE EXAMPLES
8!=8×7×6×5×4×3×2×1
A baseball manager is determining the batting order for
the team. The team has 9 players, but themanager
definitely wants the pitcher to bat last. How many batting
orders are possible?
Answer: 40,320 possible batting
orders

6. MORE EXAMPLES
5 x 8 x 2 = 80
How many outfits are possible with 5 pairs of jeans, 8 t-
shirts and 2 pairs of shoes?
Answer: 80 possible outfits

7. MORE EXAMPLES
How many outfits are possible with 5 pairs of jeans, 8 t-
shirts and 2 pairs of shoes?
Answer: 80 possible outfits

8. MORE EXAMPLES
10 x 10 x 10 x 10
How many 4 digit debit card personal identification
numbers (PIN) can be made?
Answer: 10,000 PINS
There are 10 digits. These are 0,1,2,3,4,5,6,7,8 and 9

9. PERMUTATIONS
Permutations refer to the number of ways to arrange a set
of items where the order matters. The formula for
permutations of n items taken r at a time is given by:
Example:
To find the number of ways to arrange 3 books out of 5,
use:

10. MORE EXAMPLES
In how many ways can the letters of the word “LOVE” be
arranged?
Answer: 24 ways

11. MORE EXAMPLES
In how many ways can the letters of the word “LOVE” be
arranged?
Answer: 24 ways
Listing Method

12. MORE EXAMPLES
In how many ways can you arrange five (5) people to be
seated in a row?
Answer: 120 ways

13. COMBINATIONS
Combinations refer to the number of ways to choose items
from a set where the order does not matter. The formula
for combinations of n items taken r at a time is:
Example:
To find the number of ways to choose 3 fruits from a
selection of 5, use:

14. MORE EXAMPLES
How many different committees of 4 people can be formed
from a pool of 7 people?
Therefore, the number of different committees of 4 people
that can be formed from a pool of 7 people is 35.

15. MORE EXAMPLES
Lotto is a game of chance which is played by choosing six
different numbers from 1 to 42. How many different bets are
possible?
Thus, there are 5,245,786 bets
possible.

16. Binomial probability is a
fundamental concept in
statistics and probability theory
that deals with experiments or
processes that can result in two
distinct outcomes, typically
referred to as "success" and
"failure."
BINOMIAL
PROBABILITY

18. Solution:
(a) The repeated tossing of the coin is an
example of a Bernoulli trial. According to
the problem:
Number of trials: n=5
Probability of head: p= 1/2 and hence the
probability of tail, q =1/2
For exactly two heads:
x=2
If a coin is tossed 5 times, using binomial distribution find the probability of:
(a) Exactly 2 heads
(b) At least 4 heads.
Solution:
b)
Therefore, P(x 4) = 5/32 + 1/32 =
≥
6/32 = 3/16

19. KEY POINTS TO REMEMBER
• Counting Principle is best for independent choices
across categories.
• Permutations are used when order is important.
• Combinations are used when order is not
important.
• Binomial Probability applies to situations with two
possible outcomes across multiple trials.

20. Probability helps us make better choices when we
are unsure about what might happen. By
understanding how likely different results are, we
can weigh the risks and rewards in everyday
decisions, like spending money or playing games.
Learning about probability allows us to think
clearly and make choices that fit our needs and
goals.