Here is what we have met in Problem 10.3 yesterday, about the “greatest integer less than or equal to x”. The mathematical name is “floor function”. On the other hand, we have a “ceiling function”, which is the “smallest integer greater than or equal to x”.

Please visit this website for detailed information: https://www.mathsisfun.com/sets/function-floor-ceiling.html

Lesson Review:

Geometric Probability can be used in situations where we don’t have a set of individual items to count; rather, we have infinitely many outcomes which can be nicely represented using some geometric object (a line segment, a region of a plane, etc). Or say, we use geometric probability when our outcomes are continuous rather than discrete.

We learned probability using lengths and areas. (In the future, you may meet some problems that require volumes.)

P (success) = Size of successful region/size of possible region

Homework:

• P156 Ex. 10.2.1, 10.2.3, 10.2.5
• P161 Ex 10.3.1-10.3.4
• Bonus: P162 Ex 10.7, P163 Ex 10.16

Other Resources:

https://www.khanacademy.org/math/ap-statistics/random-variables-ap/geometric-random-variable/e/geometric-probability

*Next Sunday we have no school. Wish everyone a great Thanksgiving holiday! See you on 12/08!

Lesson review：
Today we have finished the second half of Chapter 8: Basic Probability Techniques and Chapter 9: Think about it!

1. Addition or Multiplication
   We add probabilities of exclusive events: P(A or B) = P(A) + P(B);
   and multiply probabilities of independent events: P(A and B) = P(A) X P(B)
2. Independent VS Dependent Events
   Independent Events: the probability that one event occurs in no way affects the probability of the other event occurring.
   Dependent Events: the probability of one event occurring influences the likelihood of the other event.
3. Avoid unnecessary computation:
   1) take advantage of symmetry
   2) ignore the irrelevant information
   3) consider multiple events at once instead of dealing with them separately
   4) be flexible, think out of the box!

Homework:
Page 135: Ex 8.4.1, 8.4.4, 8.5.1, 8.5.4, 8.5.6
Page 151: Ex 9.8, 9.10, 9.12, 9.14

Additional Resource:
https://www.siyavula.com/read/maths/grade-11/probability/10-probability-02

https://www.wyzant.com/resources/lessons/math/statistics_and_probability/probability/further_concepts_in_probability

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb/x9e81a4f98389efdf:dependent-events-precalc/v/independent-events-1

https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/v/compound-probability-of-independent-events

See you next Sunday and I look forward to exploring the geometric probability with you! Have fun!

Flexibility Assessment:

Thanks to everyone for participating in the pilot Flexibility Assessment from Harvard Graduate School of Education. Your feedback will help to revise this assessment. This study will be carried out in Finland, Sweden, Spain, America, and China.

It’s interesting to see that most of the students in this class solved the problems in a traditional way in Part I, but developed better (more innovative) methods in Part II for the same set of problems. Perhaps because of the time constraint, or being used to the more traditional ways. We learned that for some problems, there are some “innovative strategies”. Be familiar with the properties and principles in math, you can solve problems faster, and better:)

Lesson Review:

8.2 Probabilities and Addition
If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B)
More generally, if A1, A2, …, An are mutually exclusive, then P(A1 or A2 or A3 … or An) = P(A1) + P(A2) + … + P(An)

If the events are not mutually exclusive, you have to correct for the overcounting:
P(A or B) = (P(A)+P(B)-P(A and B)
(think about the Venn Diagram)

8.3 Complementary Probabilities
P (A does not occur) = 1 – P(A occurs)

Sometimes it is easier to calculate the “opposite” events, so the complementary method can work better than the constructive method. (Refer to Example Problem 8.6)

Homework:
Page 128: 8.2.1-8.2.3, 8.2.6
Page 130: 8.3.1-8.3.4

Next week we are going to finish Chapters 8 and 9. Please be prepared and look forward to seeing you soon!

Lesson review:

1. Midterm Exam Paper Correction
2. Chapter 7: Introduction to probability
   1)P(A) = Number of outcomes for A / Number of possible outcomes
   or P(A) = Number of successful outcomes/Number of possible outcomes
   2). The range of P(A) is [0,1]

Homework:
Page 114: Ex 7.2.1-7.2.2
Page 117: Ex 7.3.1 -7.3.3
Page 120: Ex 7.4.1- 7.4.5
Page 121: Ex 7.12 (Classwork)

Annabelle gave a close guess to the “Shooting star” warm-up exercise: around 80%. The accurate probability is 84%. We will discuss it in class next Sunday.

See you next week!