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 $$P(A\cup B)$$.​

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.

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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:

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

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

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

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