The document reviews concepts related to counting and probability, including the definitions of sample space, the fundamental counting principle, and probability restrictions. It presents various practice problems to apply these concepts, such as finding integers divisible by given numbers, and calculating outcomes in probability experiments like coin flips and dice rolls. Additionally, it includes examples of different choices when selecting computer components using the counting principle.

1. Topic 5.
Counting
and
Probability
WEEK 5. AUG.2

2. Homework Review

3. Homework Review

4. Homework Review

5. Homework Review

6. Homework Review
How many positive integers less than 200 are divisible
by 2, 3 and 5?
● 6 positive integers
List the factors for 81
● 1, 3, 9, 27, 81

7. Definition of Sample Space
The sample space of an experiment or random trial
is the set of all possible outcomes or results of that
experiment.
For example, if you roll a dice, 6 things could
happen. You could roll a 1, 2, 3, 4, 5, or 6. Therefore,
1, 2, 3, 4, 5, and 6 would be the sample space for
this experiment.

8. Tree Diagram
As we all know, you can have two possible outcomes when flipping a
coin 1 time: heads or tails. But how would we find the possible outcomes
when we flip a coin 3 times?
We can find the
sample space
using a tree
diagram:

9. Listing the possibilities:
Another way to find the possible outcomes is to list them. Say we have two
dices and we roll them at the same time. What are all the possible results that
we can get?
We can find the
sample space
using a chart:

10. Practice Problem

11. Answer

12. Definition of Counting Principle
The Fundamental Counting Principle is a way to
figure out the number of outcomes in a probability
problem. Basically, you multiply the events together
to get the total number of outcomes

13. The Counting Principle
...
Question: How many different possible breakfast
combinations are there if you order one from each
group:
EGG GROUP ( Scrambled eggs, Fried Eggs,
Omelette )
MEAT GROUP ( Bacon, Sausage )

14. Answer:
...
Since there are 3
possible egg
choices and 2
possible meat
choices, we can
apply the
counting principle
and do 3 x 2 = 6

15. Practice Problem:
...

16. Answer:
...

17. Practice Problems (you don’t have to give
an exact answer, showing the equation is
enough):
...

18. Answer:
30 x 29 x 28 combinations possible
24 x 23 x 22 x 21 x 20 choices

19. Practice Problem

20. Answer

21. Definition of Probability
Probability is the extent to which something is
probable; the likelihood of something happening or
being the case.

22. Finding The Probability
Question: What is the probability of getting a head when
flipping a coin one time?

23. Practice Problems:
What is the probability of the following happening?
1. Randomly selecting the letter A in Cheatham?
1. Choosing a blue marble out of a bag filled with 3
blue marbles, 4 red marbles and 2 yellow
marbles?
1. Throwing a fair coin 2 times and getting 2 heads?

24. Answer:
What is the probability of the following happening?
1. 2/8, or 1/4
1. 3/9, or or 1/3
1. The total outcomes of throwing 2 heads: 2 x 2 = 4
The sample space of these 4 outcomes:
(HH, HT, TH, TT)
Therefore, the possibility of getting two heads (HH)
is 1/4

25. Probability Restrictions
The probability of an event occurring must be
between 0 and 1, inclusive.
An event that is for sure to happen will have a
probability of 1.
An event that cannot happen will have a
probability of 0.
The sum of the probabilities for all possible outcomes
is equal to 1.

26. Practice Problems:
What is the probability of the following happening?
1. Picking a candy that is not red out of a bag filled
with 5 blue candies, 6 purple candies and 4 red
candies?
2. Rolling a six-sided die and getting a number over
2?
3. Rolling two six-sided dice and getting a 3 and a
6?

27. Answer
What is the probability of the following happening?
1. 11/15
2. 2/3
3. 1/18

28. Review Problem:

29. Answer

30. Review Problem
To buy a computer system, a customer can choose
one of 4 monitors, one of 2 keyboards, one of 4
computers and one of 3 printers. Determine the
number of possible systems that a customer can
choose from.

31. Answer
A customer can choose one monitor, one keyboard, one
computer and one printer. The diagram below shows each
item with the number of choices the customer has.
Using the counting principle, the number of all possible
choices can be found by 4 × 2 × 4 × 3 = 96

32. Time for handouts!
Take the handouts that you printed out at home and using the
concepts that we learned today, start working on them. If you don’t
finish that’s okay, you can finish it at home.