Math 1320 Chapter 7: Probability 7.2 Relative Frequency

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.

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.

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: 240 400 =0.6

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.

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

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 = 1200+1050+1300 4400 =0.80681818181… My answer is 0.8068, rounded to four decimal places. Outcome HH HT TH TT Frequency 1200 1050 1300 850

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 2008. The four categories are mutually exclusive. Mortgage Status Current Past Due In Foreclosure Repossessed Frequency 115 70 60 5

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.

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 = 115 250 =0.46, Past Due = 70 250 =0.28, In Foreclosure = 60 250 =0.24, Repossessed = 5 250 =0.02

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 = 115 250 + 70 250 = 185 250 =0.74

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

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

An author is established and successful. An author is a new author. Answer: 56 200 =0.28 An author is a new author. Answer: 54 200 =0.27 An author is successful. Answer: 72 200 =0.36

A successful author is established Answer: 56 72 = 7 9 =0.78 An established author is successful Answer: 56 146 = 28 173 =0.38