1. Math 1320 Chapter 7: Probability
7.2 Relative Frequency

2. Definitions
When an experiment is performed a number of times, the estimated probability or relative frequency of an event E is the fraction of times that the event E occurs.
If the experiment is performed N times and the event E occurs 𝑓𝑟(𝐸) times, then the estimated probability is given by 𝑃 𝐸 = 𝑓𝑟(𝐸) 𝑁
The number 𝑓𝑟(𝐸) is called the frequency of E, N is the number of times that the experiment is performed and is called the number of trials or sample size.

3. Example
Four hundred adults are polled and 160 of them support universal health care coverage. E is the event that an adult does not support universal health coverage. Calculate the relative frequency.

4. Example
Four hundred adults are polled and 160 of them support universal health care coverage. E is the event that an adult does not support universal health coverage. Calculate the relative frequency. Solution: =0.6

5. Frequency Distributions
Let 𝑆={ 𝑠 1 , 𝑠 2 ,…, 𝑠 𝑛 } be a sample space and let 𝑃( 𝑠 𝑖 ) be the estimated probability of the event { 𝑠 𝑖 }. Then
0≤𝑃( 𝑠 𝑖 )≤1
The relative frequency of each outcome is a number between 0 and 1.
𝑃 𝑠 1 +𝑃 𝑠 2 +…+𝑃 𝑠 𝑛 =1
The relative frequencies of all the outcomes add up to 1.
If 𝐸={ 𝑒 1 , 𝑒 2 ,…, 𝑒 𝑟 }, then 𝑃 𝑒 1 +𝑃 𝑒 2 +…+𝑃 𝑒 𝑟 =𝑃(𝐸).
The relative frequency of an event E is the sum of the relative frequencies of the individual outcomes in E.

6. Example
The following table shows the frequency of outcomes when two distinguishable coins were tossed 4400 times and the uppermost faces were observed. What is the relative frequency that heads comes up at least once? (Round your answer to four decimal places).
Outcome HH HT TH TT
Frequency
1200
1050
1300
850

7. Example Solution
Two distinguishable coins were tossed 4400 times. The relative frequency that heads comes up at least once will include HH, HT and TH. So 𝑃 𝐸 = = … My answer is , rounded to four decimal places.
Outcome HH HT TH TT
Frequency
1200
1050
1300
850

8. Example (like 7&8)
The following chart shows the results of a survey of the status of subprime home mortgages in a certain state in November The four categories are mutually exclusive.
Mortgage Status
Current
Past Due
In Foreclosure
Repossessed
Frequency
115 70 60 5

9. Find the relative frequency distribution for the experiment of randomly selecting a subprime mortgage in the state and determining its status. Round to two decimal places.
What is the relative frequency that a randomly selected subprime mortgage in the state was neither in foreclosure nor repossessed? Round to two decimal places.

10. Find the relative frequency distribution for the experiment of randomly selecting a subprime mortgage in the state and determining its status. Round to two decimal places.
Solution: We need to know total mortgages first; this is 250. Current = =0.46, Past Due = =0.28, In Foreclosure = =0.24, Repossessed = =0.02

11. What is the relative frequency that a randomly selected subprime mortgage in the state was neither in foreclosure nor repossessed? Round to two decimal places.
Solution: Not in foreclosure and not repossessed means that it is current or past due. That frequency is 𝑃 𝐸 = = =0.74

12. Example
The following table shows the results of a survey of 200 authors by a publishing company. Compute the relative frequency of the given event if an author as specified is chosen at random.
New Authors
Established Authors
Total
Successful 16 56 72
Unsuccessful 38 90
128 54
146
200

13. An author is established and successful.
An author is a new author.
An author is successful.
A successful author is established
An established author is successful

14. An author is established and successful. An author is a new author.
Answer: =0.28
An author is a new author.
Answer: =0.27
An author is successful.
Answer: =0.36

15. A successful author is established
Answer: = 7 9 =0.78
An established author is successful
Answer: = =0.38