## 7.2: Relative Frequency (Estimated Probability)

To start, here are some basic definitions.
 Definition Example The frequency of the event E is the number of times the event E occurs. fr(E) = the number of times E occurs Toss a coin 20 times. If heads comes up 13 times, then the frequency of the event that heads comes up is fr(E) = 13. The relative frequency or estimated probability of the event E is the fraction of times E occurs. P(E) = \frac{fr(E)}{N} Note: It follows that P(E) must be a number between 0 and 1 (inclusive). Referring to the situation above, the relative frequency of the event that heads comes up is P(E) = \frac{fr(E)}{N} = \frac{13}{20} The number of times that the experiment is performed is called the sample size. N = number of times the experiment is performed. The experiment above was performed 20 times, so this is the sample size; N = 20.

Note: If E happens to consist of a single outcome s, then we refer to P(E) as the relative frequency of the outcome s, and we write P(s).

A pair of dice (one red, one green) is cast 30 times, and on 4 of these occasions, the sum of the numbers facing up is 7. Let E be the event that the sum is 7.
 Q The relative frequency that the sum is 7 is P(E) = . Answer should be accurate to 4 decimal places.

Dice Simulation
To simulate the above experiment, press the "Throw Dice" button 30 times, or press the "Throw Dice 10 x" button three times. E is the event that the sum is 7, and its relative frequency will be calculated for you.  N (Sample size) Sum (Red + Green) fr(E) (Frequency) P(E) (relative frequency)

You will see in the next tutorial that the modeled probability of E is 1/6 = .1666... The relative frequency should approach this number as the sample size gets large. You can now verify this experimentally.

In 1993, there were approximately 10,000 fast food outlets in the US that specialized in Mexican food. Of these, the largest were Taco Bell with 4809 outlets, Taco John's with 430 outlets and Del Taco with 275 outlets.* The relative frequency that a fast food outlet that specializes in Mexican food is none of the above is:
 Q P(E) = Answer should be accurate to 4 decimal places.

* Source: Technomic Inc./The New York Times, February 9, 1995, p. D4.

You can find more examples similar to those above in Section 7.2 of and .

Relative Frequency Distribution

The collection of the relative frequencies of all the outcomes is the relative frequency distribution or estimated probability distribution.

Example
If 10 rolls of a die resulted in the outcomes 2, 1, 4, 4, 5, 6, 1, 2, 2, 1, then the associated relative frequency distribution is the one shown in the following table.

 Outcome s 1 2 3 4 5 6 Frequency fr(s) 3 3 0 2 1 1 Relative Frequency P(s) 0.3 0.3 0 0.2 0.1 0.1

You run a commercial website that specializes in the sale of video games. The following statistics show the number of downloads of your five on-line video games last week.

 Game Dragon Quest Star Pilot Galactic Warrior 4 Detective III Advanced Star Pilot Downloads 120 50 15 55 10

Q Fill in the following relative frequency table and press "Check."

 Outcome Dragon Quest Star Pilot Galactic Warrior 4 Detective III Advanced Star Pilot Relative Frequency     Q The estimated probability that a downloaded game is either Star Pilot or Advanced Star Pilot is .

Following are some of the properties of relative frequency. Which one did you use in answering the last question?
 Some Properties of Relative Frequency 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 (a) 0 &le P(s_i) ≤ 1 (b) P(s_1) + P(s_2) + ... + P(s_n) = 1 (c) If E = \{\.e_1, .., e_r\.\}, then P(E) = P(e_1) + ... + P(e_r). In words: (a) The relative frequency of each outcome is a number between 0 and 1. (b) The relative frequencies of all the outcomes add up to 1. (c) The relative frequency of an event E is the sum of the relative frequencies of the individual outcomes in E.

For more practice, try some of the questions in the chapter review exercises (Warning: it covers the whole of Chapter 7). Also try the exercises dealing with estimated probability in Section 7.2 of and .

Last Updated: April, 2009