"A or B": the addition rule Draw one card. What's the probability it's red or an ace?
There are \(26\) red cards and \(4\) aces, but the two red aces were counted twice.
\[P(\text{red or ace}) = \frac{26}{52} + \frac{4}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}\]
DefinitionMutually exclusive events
Events are mutually exclusive if they can't happen together: \(P(A \text{ and } B) = 0\) .\(P(A \text{ or } B) = P(A) + P(B)\) .
Conditional probability Sometimes you already know part of the result. Conditional probability updates the probability with that information.
DefinitionConditional probability
The probability of \(A\) given \(B\) is
\[P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}\]
Two-way tables make conditional probabilities easy to read: restrict attention to the row or column of the given condition.
ExampleA two-way table
A survey of \(100\) students:
Honor roll Not honor roll Total Plays a sport \(25\) \(15\) \(40\) No sport \(20\) \(40\) \(60\) Total \(45\) \(55\) \(100\)
\(P(\text{honor roll}) = \frac{45}{100} = 0.45\) .
\(P(\text{honor roll} \mid \text{sport}) = \frac{25}{40} = 0.625\) : look only at the \(40\) athletes.
Independent events DefinitionIndependent events
\(A\) and \(B\) are independent if knowing one happened doesn't change the probability of the other: \(P(A \mid B) = P(A)\) .
\[P(A \text{ and } B) = P(A) \cdot P(B)\]
In the survey, \(P(\text{honor roll} \mid \text{sport}) = 0.625 \neq 0.45 = P(\text{honor roll})\) : the events are dependent .
RuleMultiplication rule
For any events: \(P(A \text{ and } B) = P(A) \cdot P(B \mid A)\) . For independent events this becomes \(P(A) \cdot P(B)\) .
ExampleWith and without replacement
Draw two cards. What's the probability both are kings?
Without replacement (dependent):
\[\frac{4}{52} \cdot \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}\]
With replacement (independent): \(\frac{4}{52} \cdot \frac{4}{52} = \frac{1}{169}\) .
Watch outMutually exclusive isn't independent
If two events with positive probabilities are mutually exclusive, they're dependent : knowing one happened makes the other impossible.
Practice Practice
\(A\) and \(B\) are independent, with \(P(A) = 0.6\) and \(P(B) = 0.3\) .\(P(A \text{ and } B)\) and \(P(A \text{ or } B)\) .
Show Hide the solution\(P(A \text{ and } B) = 0.18\) .\(\quad P(A \text{ or } B) = 0.6 + 0.3 - 0.18 = 0.72\) .
Practice
The chance of rain is \(30\%\) each day, independently. What's the probability of no rain for three days in a row?
Show Hide the solutionPractice
Using the survey table, find \(P(\text{sport} \mid \text{honor roll})\) .
Show Hide the solutionLook only at the \(45\) honor roll students: \(\frac{25}{45} = \frac{5}{9} \approx 0.556\) .
Quick check
You roll a die and flip a coin. What is \(P(\text{even and heads})\) ?
A \(\frac{1}{2}\) B \(\frac{1}{4}\) C \(1\) D \(\frac{1}{12}\)
Independent events: \(\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\) .
Quick check
Which statement means \(A\) and \(B\) are independent?
A \(P(A \text{ and } B) = 0\) B \(P(A \mid B) = P(A)\) C \(P(A) + P(B) = 1\) D \(P(A \text{ or } B) = P(A) + P(B)\)
Knowing \(B\) happened doesn't change the probability of \(A\) .