1. An Introduction to…

2. chance
likelihood 1

3. equally likely unlikely impossible very unlikely likely certain
very likely

6. Probability of Events P(face card) ? 𝑃 𝐹𝐶 = 𝑛(𝐹𝐶) 𝑛(𝑆) 𝑃 𝐹𝐶 = 12 52
= 3 13

7. Rules of Probability Complement of an Event
The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A. 7

8. 1
1 – P(E)

10. Rules of Probability Union of Two Events
(Figure 5.5)
The union of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A  B or “A or B”).
 may be read as “or” since one or the other or both events may occur. 10

11. Rules of Probability Intersection of Two Events
The intersection of two events A and B (denoted A  B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B.
 may be read as “and” since both events occur. This is a joint probability. 11

12. Example
* First, draw a Venn diagram with Set A being even numbers, and Set B being prime numbers.

14. 𝑃 𝐴 = 2 6
𝑃 𝐴′ = 4 6
𝑃 𝐴 +𝑃 𝐴′ = 6 6
𝑃 𝐴 +𝑃 𝐴′ =1
𝑃 𝐴′ =1−𝑃(𝐴)
see next slide…

15. 𝑃 𝐴 =0.7 𝑃 𝐵 =0.5 𝑃 𝐴∪𝐵 =0.9 𝑃 𝐴∩𝐵 =0.3 𝑃 𝐴 +𝑃 𝐵 =1.2 ?
𝑃 𝐴 +𝑃 𝐵 = ?
* Impossible:  this is greater than 𝐴∪𝐵  total probability cannot exceed 1
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃 𝐴∩𝐵
CHECK:
0.9 = −0.3

16. Rules of Probability General Law of Addition
The general law of addition states that the probability of the union of two events A and B is:
P(A  B) = P(A) + P(B) – P(A  B)
So, you have to subtract P(A  B) to avoid over-stating the probability.
When you add the P(A) and P(B) together, you count the P(A and B) twice.
A and B A B

17. Events that are NOT Mutually Exclusive
These events are not mutually exclusive because two of the 4 aces are also red cards so they would be counted TWICE if you added all the red cards + all the aces…
Consider the problem of taking a card at random from a pack and you are wondering what is the chance you will get a red card or an ace?

18. Events that are NOT Mutually Exclusive
P(red OR ace) = P(red) + P(ace) = 26/52 + 4/52 = 30/52
* This is INCORRECT ! What is the true probability of getting a red card OR an ace?
P(red or ace) = 28/52

20. Practice !
You are now ready for some practice! Try page 354, Exercise 8J