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Conditional probability

Probability formulae

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Normal distribution

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The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

Conditional probability

Conditional probability in Venn diagrams

Probability formulae

Conditional probability: Tree diagrams

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Probability: Tree diagrams

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Explainer Video

Tutor: Alice

Summary

Probability formulae

In a nutshell

Often, you will find that it isn't practical to draw a Venn diagram for each problem. Instead, there are some formulae you can use to solve problems involving probabilities.

The addition formula

The addition formula can help to calculate $P(A \cap B)$ or $P(A\cup B)$ .

{P(A\cup B)=P(A)+P(B)-P(A\cap B)}

$P(A)$	Probability of event $A$ .
$P(B)$	Probability of event $B$ .
${P(A\cap B)}$	Probability of the intersection between events. Probability of $A$ and $B$ .
${P(A\cup B)}$	Probability of the union between events. Probability of $A$ or $B$ or both.

Example 1

Given that $P(A)=0.4$ , $P(B)=0.2$ and $P(A\cup B)=0.5$ , find $P(A\cap B)$ .

Rearrange the addition formula, and substitute values to find $P(A \cap B)$ :

$\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\ P(A\cap B)&=P(A) + P(B)-P(A\cup B)\\ P(A\cap B)&=0.4+0.2-0.5\\P(A\cap B)&\underline{=0.1}\end{aligned}$ 

Conditional probability

There is also a formula used to calculate conditional probabilities.

{P(A|B)=\dfrac{P(A\cap B)}{P(B)}}

$P(A)$	Probability of event $A$ .
$P(B)$	Probability of event $B$ .
${P(A\cap B)}$	Probability of the intersection between events .
$P(A\| B)$	Probability of $A$ occurring given that $B$ has occurred.

Example 2

In a class of $30$ students, $20$ are brunette and, of those $20$ , $18$ are brown eyed.

If a student is chosen at random, what is the probability of him being brown eyed (A) given that he's brunette (B)?

You can find:

$P(B)=\dfrac{20}{30}=\dfrac{2}{3}$

$P(A\cap B)=\dfrac{18}{30}=\dfrac{3}{5}$

$P(A|B)=\dfrac{\frac{3}{5}}{\frac{2}{3}}=\underline{\dfrac{9}{10}}$

Learn with Basics

Length:

Unit 1

Basics of probability

Unit 2

Calculating probabilities

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Optional

Unit 3

Probability formulae

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FAQs - Frequently Asked Questions

Is the formula P(AUB)=P(A)+P(B) correct?

Only if the events are mutually exclusive.

How can the conditional probability P(A|B) be calculated if A and B are independent?

P(A|B) will be the same as (A) if A and B are independent.

How can you find the probability of the union of two events?

By adding the probability of each event and subtracting the probability of its intersection.