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Classical probability

P(A) = \dfrac{\text{number of favorable outcomes}}{\text{total number of outcomes}}

This assumes every outcome is equally likely — check that assumption before applying the formula.

Conditional probability

The probability of $A$ given that $B$ has already happened:

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

Independence

Events $A$ and $B$ are independent exactly when knowing one doesn't change the probability of the other:

P(A \cap B) = P(A)\cdot P(B) \qquad\Longleftrightarrow\qquad P(A\mid B) = P(A)

This is a completely different idea from being mutually exclusive (unable to happen together, $P(A\cap B)=0$ ). In fact, two events with positive individual probabilities can never be both independent and mutually exclusive at once — if they can't happen together, knowing one happened tells you a lot about the other (namely, that it didn't), which is the opposite of independence.

Worked example

Problem: A bag has 4 red and 6 blue balls. Two balls are drawn without replacement. Find the probability both are red.

Solution (direct count): $P = \dfrac{\binom42}{\binom{10}{2}} = \dfrac{6}{45} = \dfrac{2}{15}$ .

Solution (sequential/conditional): $P(\text{1st red}) = \tfrac{4}{10}$ . Given the first was red, 3 red remain out of 9 total: $P(\text{2nd red} \mid \text{1st red}) = \tfrac39$ .

P(\text{both red}) = \dfrac{4}{10}\cdot\dfrac{3}{9} = \dfrac{12}{90} = \dfrac{2}{15}

Both methods agree, as they must — this is a useful way to check your work on drawing-without-replacement problems.

Practice problems

1.A fair die is rolled. Find the probability of rolling a number greater than 4.

2.Two fair coins are flipped. Find the probability both show heads, given that at least one shows heads.

3.A bag has 5 white and 3 black marbles. Two are drawn without replacement. Find the probability both are black.

4.If $P(A) = 0.4$ , $P(B)=0.5$ , and $A, B$ are independent, find $P(A \cap B)$ and $P(A \cup B)$ .

Probability Basics

Classical probability

Conditional probability

Independence

Practice problems