Simulations and Simple Probability

 1. The probability scale Thang đo xác suất.

Probability is a measure of the likely occurrence of an event.

Xác suất là thước đo mức độ có khả năng xảy ra của một sự kiện.

The probability of an event occurring is measured with a scale ranging from and including 0 (impossible event) to 1 (certain event).

Xác suất xảy ra của một sự kiện đuợc đo bằng thang điểm từ 0 (sự kiện không thể xảy ra) đến 1 (sự kiện chắc chắn xảy ra).

2. Sample spaces Không gian mẫu

- An outcome is a particular result of an experiment.

Kết quả là một kết quả cụ thể của một thí nghiệm.

- A sample space is a list of all possible outcomes.

Không gian mẫu là danh sách tất cả các kết quả có thể xảy ra.

3. Calculating theoretical probability Tính xác suất lý thuyết.

- A favourable outcome is an outcome that you want or are looking for.

Một kết quả thuận lợi là một kết quả mà bạn muốn hoặc đang tìm kiếm.

- Equally likely outcomes have the same chance of occurring.

Các kết quả có khả năng xảy ra như nhau có cùng cơ hội xảy ra.

Exercises

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4. The complement of an event

The probability of the complement of an event is the probability that the event will not happen.

For example, if an event is ‘it will rain today’, then the complement

of this is ‘it will not rain today’. The other way to interpret the

complement is to recognise it as the opposite of the event.

If the probability of an event is denoted by Pr(R), then the probability of the complement is denoted by Pr(R′).

In order to calculate Pr(R′) it is necessary to understand that the sum of the probability of an event and its complement is equal to 1:

                                   Pr(R) + Pr(R′) = 1

This equation can be manipulated so it is expressed in terms of the complement:              Pr(R′) = 1 − Pr(R)

For example, if the probability of rain today is 7/10, then:

Pr(R′) = 1 - 7/10 = 3/10

This means there is a 30% chance that it will not rain.

Exercise

5. Multi-stage experiments

Determine the probability of multi-stage experiments using tables, tree diagrams and arrays.

   5.1. Arrays

- When two events occur to form a single result, an array can be used to display the sample space.

Example 1: 

Consider an experiment in which a red die and a blue die are rolled, and the two numbers that land on top are recorded. The array shows the sample space for that experiment. There are 36 possible outcomes in this sample space.

- An array is also useful when there are two events, but a different number of outcomes for each event

Example 2: 

The following array shows the sample space for an experiment in which a coin is flipped and a die is rolled.

Exercises

Q12. 

Q13. 

A spinner is divided into 5 sections (red, blue, green, yellow and orange). A six-sided die is also rolled.

Draw the array for one spin of the spinner and one roll of the die.

Q14. 

Q15. 

Q16. 

Q17. 

Q18. 

Q19. 

Q20. 

Q21. 

Q22. 

Q23. 

Q24.

Q25.

Q26.