1. 13.4 – Compound Probability
Chapter 13 – Probability
13.4 – Compound Probability

2. Compound Events
A compound event is an event that is made up of 2 or more events.
2 Types:
Independent Events – if the occurrence of one event does not affect how the other event occurs
Dependent Events – if the occurrence of one event does affect how the other event occurs.

3. does not Independent Events
Events where the occurrence of one of the events ______ _____ affect the occurrence of the other event.
does not
A and B are independent if and only if:
P(A ∩ B) = P(A) × P(B)
P(A and B) = P(A) × P(B)
“and” → multiplication
P(A and B and C) = P(A) × P(B) × P(C)

4. Independent Events
If one student in the class was born on June 1st can another student also be born on June 1st?
If you roll a die and get a 6, can you flip a coin and get tails?

5. Independent Events P(A and B) = P(A) × P(B) P(A ∩ B) = P(A) × P(B)
Practice: A coin and a die are tossed simultaneously. Determine the probability of getting heads and a 3.
P(A and B) = P(A) × P(B)
P(A ∩ B) = P(A) × P(B)
P(H and 3) = P(H) × P(3) = 𝟏 𝟐 x 𝟏 𝟔 = 𝟏 𝟏𝟐

6. P(B and B) = P(B) × P(B) = 𝟗 𝟏𝟓 x 𝟗 𝟏𝟓 = 𝟖𝟏 𝟐𝟐𝟓
Independent Events
Practice: There are 9 brown boxes and 6 red boxes on a shelf. Anna chooses a box and replaces it. Brian does the same thing. What is the probability that Anna and Brian choose a brown box?
P(B and B) = P(B) × P(B) = 𝟗 𝟏𝟓 x 𝟗 𝟏𝟓 = 𝟖𝟏 𝟐𝟐𝟓

7. Independent Events Practice P (A) = 2 9 P (B) = 3 5
and P(A and B) = 2 15
Are A and B independent events? Explain your answer.
A and B are independent if P(A and B) = P(A) × P(B).
Independent events
𝟐 𝟗 x 𝟑 𝟓 = 𝟔 𝟒𝟓 = 𝟐 𝟏𝟓 , Thus they are independent

8. Mutually Exclusive Events
A bag of candy contains 12 red candies and 8 yellow candies.
Can you select one candy that is both red and yellow?

9. Mutually Exclusive Events
Two events, A and B are mutually exclusive if whenever A occurs it is impossible for B to occur and vice-versa.
Events such as A and A’ must be mutually exclusive.
If events A and B are mutually exclusive, then A ∩ B = ∅
Events A and B are mutually exclusive if and only if P(A and B) = 0 or P(A ∩ B) = 0

10. Mutually Exclusive Events
Please Note: If events are mutually exclusive, then
the probability of either one event or the other event
occurring is given by:
P(either A or B) = P(A) + P(B)

11. Mutually Exclusive Events
Practice: Of the 31 people on a bus tour, 7 were born in Scotland and 5 were born in Wales.
Are these events mutually exclusive?
If a person is chosen at random, find the probability that he or she was born in:
Scotland
Wales
Scotland or Wales
Yes, what is the probability of being born in Scotland AND Wales? Zero!
P(S) = 𝟕 𝟑𝟏
Mutually exclusive events
P(W) = 𝟓 𝟑𝟏
P(S or W) = P(S) + P(W) = 𝟕 𝟑𝟏 x 𝟓 𝟑𝟏 = 𝟏𝟐 𝟑𝟏

12. P(A or B) = P(A) + P(B) – P(A and B) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Overlapping Events
Overlapping Events or Combined Events ~Have outcomes that are in common~ Like the example of drawing a number 1 through 20 that is a multiple of 3 or 5. Drawing the number 15 would fit both of these outcomes
For overlapping events:
P(A or B) = P(A) + P(B) – P(A and B)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

13. Overlapping Events Solution:
Example:
50 people are members at a gym. 28 of those people play tennis. 14 of those people play basketball and 7 play both tennis and basketball. What is the probability that a randomly selected person plays tennis or basketball?
Solution:
P(T) = , P(B) = , P(T ∩ B) = 7 50
= =

14. Overlapping Events Practice 100 people were surveyed:
72 people have had a beach holiday
16 have had a skiing holiday
12 have had both
What is the probability that a person chosen has had a beach holiday or a ski holiday?
Solution:
P(B) = , P(S) = , P(T and B) =
= = 0.74

15. If P(A) = 0.6 and P(A or B) = 0.7 and P(A and B) = 0.3, find P(B).
Overlapping Events
Practice
If P(A) = 0.6 and P(A or B) = 0.7 and P(A and B) = 0.3, find P(B).
Solution:
P(A or B) = P(A) + P(B) – P(A and B)
P(B) = P(A or B) – P(A) + P(A and B)
P(B) = = 0.4
Combined events (inclusive events)

16. There are 9 brown boxes and 6 red boxes on a shelf.
Dependent Events
There are 9 brown boxes and 6 red boxes on a shelf.
What if Anna choose the box and did not replace it?
Then Brian’s event of choosing a box becomes dependent.
If Anna chooses red,
P(Brian chooses brown) =
If Anna chooses brown,
P(Brian chooses brown) =
P(Anna then Brian choose brown)

17. Dependent Events
Events where the occurrence of one of the events ______ affect the occurrence of the other event.
does
P(A then B) = P(A) × P(B given that A has occurred)

18. Dependent Events P(A then B) = P(A) × P(B given that A has occurred)
Practice: A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one by one from the box, without replacement. Find the probability that: (a) both are red (b) the first is red and the second is yellow.
P(A then B) = P(A) × P(B given that A has occurred)
P(2R) = 𝟒 𝟔 x 𝟑 𝟓 = 𝟏𝟐 𝟑𝟎 = 𝟐 𝟓
P(RY) = 𝟒 𝟔 x 𝟐 𝟓 = 𝟖 𝟑𝟎 = 𝟒 𝟏𝟓

19. Dependent Events Prime Numbers = {2,3,5,7,11,13,17,19}
Practice: A hat contains tickets with numbers 1, 2, 3, … , 19, 20 printed on them. If 3 tickets are draw from the hat, without replacement, determine the probability that all are prime numbers.
Prime Numbers = {2,3,5,7,11,13,17,19}
P(3P) = 𝟖 𝟐𝟎 x 𝟕 𝟏𝟗 x 𝟔 𝟏𝟖 = 𝟑𝟑𝟔 𝟔𝟖𝟒𝟎 =

20. Laws of Probability Type Definition Formula Mutually Exclusive Events
events that cannot happen at the same time
P(A and B) = 0 / P(A ∩ B) = 0
P(A or B) = P(A) + P(B)
P(A  B) = P(A) + P(B)
Overlapping Events
events that can happen at the same time
P(A or B) = P(A) + P(B) – P(A and B)
P(AB) = P(A) + P(B) – P(A∩B)
Independent Events
occurrence of one event does NOT affect the occurrence of the other
P(A and B) = P(A) • P(B)
P(A ∩ B) = P(A) • P(B)
Dependent Events
occurrence of one DOES affect the occurrence of the other
P(A then B) = P(A) • P(B)

21. A sample space is the set of all possible outcomes of an experiment.
4 ways to create a Sample Set:
List the outcomes {you must use curly brackets}
Table of outcomes
Draw a 2 Dimensional Grid
Draw a Tree Diagram

22. Sample Space List the outcomes for: Practice a) Flipping a coin
b) Rolling a single die
c) Flipping two coins
d) Rolling two dice

23. Sample Space
Practice
Use a 2D grid to illustrate the sample space for tossing a coin and rolling a die simultaneously. Find the probability of:
Rolling a Die 1 2 3 4 5 6
Coin Toss H H1 H2 H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6
a) tossing heads on the coin b) getting tails and a 5 c) getting tails or a 5

24. Sample Space
Practice
Two square spinners, each with 1, 2, 3, and 4 on their edges, are twirled simultaneously. Draw a 2D grid of the possible outcomes. Find the probability of:
Spin 2 1 2 3 4
Spin 1
(1,1)
(1,2)
(1,3)
(1,4)
(2,1)
(2,2)
(2,3)
(2,4)
(3,1)
(3,2)
(3,3)
(3,4)
(4,1)
(4,2)
(4,3)
(4,4)
a) getting a 3 with each spinner b) getting a 3 and a 1 c) getting an even result for each spinner

25. Sample Space
Practice: A red and blue dice are rolled together. Calculate the probability that: (a) The total score is 7 (b) The same number comes up on both dice (c) The difference between the scores is 1 (d) The score on the red dice is less than the score on the blue dice (e) The total score is a prime number
Roll 2 1 2 3 4 5 6
Roll 1
(1,1) 2
(1,2) 3
(1,3) 4
(1,4) 5
(1,5) 6
(1,6) 7
(2,1) 3
(2,2) 4
(2,3) 5
(2,4) 6
(2,5) 7
(2,6) 8
(3,1) 4
(3,2) 5
(3,3) 6
(3,4) 7
(3,5) 8
(3,6) 9
(4,1) 5
(4,2) 6
(4,3) 7
(4,4) 8
(4,5) 9
(4,6) 10
(5,1) 6
(5,2) 7
(5,3) 8
(5,4) 9
(5,5) 10
(5,6) 11
(6,1) 7
(6,2) 8
(6,3) 9
(6,4) 10
(6,5) 11
(6,6) 12
Example 9 from page 503 of 2nd edition

26. Sample Space
Practice
Draw a table of outcomes to display the possible results when two dice are rolled and the scores are summed. Determine the probability that the sum of the dice is 7.
Roll 2 1 2 3 4 5 6
Roll 1 7 8 9 10 11 12
Example 9 from page 503 of 2nd edition