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Calculating probabilities

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Explainer Video

Tutor: Bilal

Summary

Calculating probabilities

In a nutshell

Probabilities describe the likelihood of an event occurring. They are quantities that are always between $0$ and $1$ .

Definitions

Here are some definitions that you need to know.

term	definition
Experiment	A repeatable process.
Event	One or more possible outcomes of an experiment.
Sample space	The set of all possible outcomes of an experiment.

Representing sample space

The sample space of an experiment can be represented with a sample space diagram. These are an easy way to list possible outcomes.

Example 1

A fair coin is tossed $3$ times. What is the sample space of this experiment?

Let $H$ represent heads, and $T$ represent tails.

The set of possible options is:

$\underline{\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}}$

Example 2

Two fair dice are tossed, and the numbers that they land on are multiplied together. What is the probability that the two numbers multiply to give $12$ ?

Draw a table to represent the sample space:

	die 1
die 2	$\bf \times$	$\bf1$	$\bf2$	$\bf3$	$\bf4$	$\bf5$	$\bf6$
	$\bf1$	$1$	$2$	$3$	$4$	$5$	$6$
	$\bf2$	$2$	$4$	$6$	$8$	$10$	$12$
	$\bf3$	$3$	$6$	$9$	$12$	$15$	$18$
	$\bf4$	$4$	$8$	$12$	$16$	$20$	$24$
	$\bf5$	$5$	$10$	$15$	$20$	$25$	$30$
	$\bf6$	$6$	$12$	$18$	$24$	$30$	$36$

Identify the desired events in the sample space:

	die 1
die 2	$\bf \times$	$\bf1$	$\bf2$	$\bf3$	$\bf4$	$\bf5$	$\bf6$
	$\bf1$	$1$	$2$	$3$	$4$	$5$	$6$
	$\bf2$	$2$	$4$	$6$	$8$	$10$	$\large\textcircled{\small12}$
	$\bf3$	$3$	$6$	$9$	$\large\textcircled{\small12}$	$15$	$18$
	$\bf4$	$4$	$8$	$\large\textcircled{\small12}$	$16$	$20$	$24$
	$\bf5$	$5$	$10$	$15$	$20$	$25$	$30$
	$\bf6$	$6$	$\large\textcircled{\small12}$	$18$	$24$	$30$	$36$

Calculate the probability:

$4$ outcomes out of $6\times6=36$ options.

The probability that the two numbers multiply to give $12$ is $\dfrac{4}{36}=\underline{\dfrac{1}{9}}$ .

Learn with Basics

Length:

Unit 1

Calculating probability

Unit 2

Basics of probability

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Unit 3