Probability under Frequency Distribution

The theory of probability grew out of the various games of chance using coins, dice, and cards. Since they lend themselves well to the application of concepts of probability, they will be used in probability as a topic, but for now we will be learning how to answer probability questions under frequency distribution. Before that, the below basics concepts needed to be understood:

Processes such as flipping a coin, rolling a die, or drawing a card from a deck are called probability experiments. Probability experiment is a chance process that leads to well-defined results called outcomes.

An Outcome is the result of a single trial of a probability experiment.

A Sample Space is the set of all possible outcomes of a probability experiment. That is in an experiment, the set of all possible outcomes is called the sample space.

An Event consists of a set of outcomes of a probability experiment. An event can be one outcome or more than one outcome.

An event with one outcome is called a sample event. For example, if a die is rolled and a 6 shows, this result is called an outcome, since it is a result of a single trial.

The event of getting an odd number when a die is rolled is called a compound event, since it consists of three outcomes or three sample events. In general, a compound event consists of two or more outcomes or sample events.

Consider a sample space which is made up of Z equally likely outcomes. If the event E can happen in Y ways out of the total Z, then we say

NB: Probability under frequency distribution is calculated using the frequencies only.

ü  At least, means exactly or more than. Example at least 5, means exactly 5 or more than 5, i.e. {5, 6, 7, 8, 9 ...}

ü  At most means exactly or less than. Example at most 7, means exactly 7 or less than 7, i.e. {... 3,  4, 5, 6, 7}

ü  Between two numbers say 10 and 15 means {11,  12, 13, 14}

ü  Less than a number, i.e. less than 10 means {... 7, 8, 9}

ü  Greater than a number, i.e. greater than 16 means {17, 18, 19 ...}

Example one: The distribution of marks scored in a test by a number of students in a class is shown in the table below;

Marks	10-14	15-19	20-24	25-29	30-34	35-39	40-44	45-49
Frequency	1	5	4	5	8	2	3	2

Find the probability that a student chosen random from the class scored

a.       a mark between 19 and 40

b.      at least 35 marks

c.       at most 34 marks

d.      a mark in the modal class

a.       a mark in the median class

b.      a mark of 40 and above

c.       a mark below 30

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