This document covers key concepts in probability, including sample spaces, events, rules of probability such as addition and multiplication, and conditional probability. It defines probability as the proportion of times an outcome occurs in a long series of repetitions. It introduces terminology like the sample space, events, and rules for calculating probabilities of individual events and combined events, depending on whether they are disjoint, independent, or conditional. Several examples demonstrate how to apply the rules of probability, addition, multiplication, and conditional probability to calculate probabilities in finite sample spaces.

1. Probability: The Study of
Randomness
AP Statistics, 11th Grade
Miss Salma

2. Lesson Objectives and Standards
Lesson Objectives:
SWBAT use the rules of probability and conditional probability including those for addition,
multiplication, and complementation to solve probabilities in finite sample spaces.
Standards:
1.0 Students solve probability problems with finite sample spaces by using the rules for addition,
multiplication, and complementation for probability distributions and understand the
simplifications that arise with independent events.
2.0 Students know the definition of conditional probability and use it to solve for probabilities in
finite sample spaces.

3. Where Can We Find Probability?
❖ Probability is everywhere!
Here is a great example of
where you can find
probability. It’s literally a
tower bursting of
probability.

4. Randomness and Probability
❖ A phenomenon is random if individual outcomes are uncertain but there
is still a regular distribution of outcomes in a large number of
repetitions.
❖ The probability of any outcome of a random process is the proportion of
times the outcome would occur in a very long series of repetitions (Law
of Large Numbers)

5. Probability Terminology
❖ The sample space, S, of a random
phenomenon is the set of all possible
outcomes. (heads or tails)
❖ An event (A, B, C) is an outcome or set of
outcomes of a random phenomenon. An
event is a subset of the sample space.
(heads)
❖ The probability of Event A, where all
outcomes are equally likely:
❖ P(A)=(count of outcomes in A)/(count of
outcomes in S)

6. Now You Try: Example 1
Example 1a. What is the probability of
heads if we flip a coin?
Example 1b. What is the probability of
rolling an even number on a six-sided
die?

7. Rules of Probability
1.0 ≤ P(A) ≤ 1
2.P(S) = 1
3.If P(A) = 1, event A always occurs. (certain)
4.If P(A) = 0, event A never occurs. (impossible)
5.P(A) + P(Not A) =1
Not A or Ā is called the complement of A.

8. Addition Rule For Disjoint Events
Two events A and B are disjoint if they have no outcomes
in common, therefore can never occur together.
Disjoint ↔ P(A and B) = 0
P(A or B) = P(A) + P(B)

9. Example 2
Roll an even number or a “one” on a six-sided die.
A= {2,4,6}, B{1}
P(A or B) = P(A) +P(B) → disjoint
3/6 + 1/6 = 4/6

10. General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)

11. Now You Try: Example 3
What’s the probability of rolling an even number or a number greater than 3?

12. Independence and Multiplication Rule
Two events A and B are independent if knowing
that A occurs doesn’t change the probability that
the B occurs.
Ex. Rolling a 2 and flipping a head.
If A and B are independent,
P(A and B) = P(A) × P(B)

13. Example 4
Suppose we toss a fair coin twice. What is the probability that both tosses are heads.
Solution:
A =first toss is heads
B = second toss is heads
S= { HH, HT, TH, TT}
Half of half
P(A and B) = P(A) × P(B)
0.5 X 0.5 = 0.25

14. General Multiplication Rule
P(A and B) = P(A) × P(B|A)
Conditional Probability: P(B|A) The probability of B
given A has occurred.

15. Independent vs. Disjoint
Disjoint ↔ P(A and B) = 0
Independent ↔ P(A and B) = P(A)∗ P(B)
Independent ↔ P(B) = P(B|A)

16. Conditional Probability
P(A and B) = P(A) × P(B|A)
P(B|A) = P(A and B)
P(A)

17. Now You Try: Example 5
Below is the distribution of U.S. college students classified by age and full-
time or part-time status.
a. What is the probability that a
student is full time?
b. What is the probability that a
student is 20 to 24 and is
full time?

18. Now You Try: Culminating Question
After watching the video, answer the
following question:
How sure are you that Jimmy has the disease?
A. He almost certainly has the disease.
B. He probably has the disease.
C. It’s about 50-50.
D. He probably does not have the disease.