A quantity that takes different values by chance in an observation or experiment is called a random variable; for example the points on a thrown die. A table listing the values of X and their probabilities is the probability distribution table; in practice we build tables of frequencies or relative frequencies and show them by a frequency polygon or a histogram. The sum of all frequencies equals the number of trials: ΣM = N, and the sum of the relative frequencies equals 1: ΣW = 1. A variable taking separate values is discrete; one that can take every value in an interval is continuous. With two dice the sum of points takes values from 2 to 12, and the most likely sum is 7 (probability 6/36 = 1/6).
Worked examples
A coin is tossed 3 times, X is the number of heads. Values 0, 1, 2, 3 have probabilities 1/8, 3/8, 3/8, 1/8; they add up to 8/8 = 1.
20 students were asked: X is the number of books read. Values 1, 2, 3 have frequencies 5, 10, 5; ΣM = 20 = N; W = 1/4, 1/2, 1/4, ΣW = 1.
Class activity
“Build the table”: the class throws two dice and records sums over 36 trials; a frequency table and polygon are made and compared with the theoretical probabilities.
Practice
1
Is the time from home to school discrete or continuous?
Continuous.
2
What kind of variable is the number of students absent per day?
Discrete.
3
If the frequencies are 3, 5, 2, what is the number of trials N?
10
4
Why do all probabilities in a distribution table add up to 1?
The table covers all possible outcomes, which together form a certain event.