Sample spaces and events, theoretical probability, complements, using counting in probability, and experimental probability with the law of large numbers.
Outcomes, sample spaces and events
A probability experiment is any action with uncertain results: rolling a die, drawing a card, flipping a coin.
Rolling a die, the sample space is \(\{1, 2, 3, 4, 5, 6\}\), and "rolling an even number" is the event \(\{2, 4, 6\}\).
Theoretical probability
Complements
The complement is often the easy way to handle "at least one".
Using counting
For larger experiments, count favorable and total outcomes with permutations or combinations.
Experimental probability
Experimental (empirical) probability is the relative frequency observed in actual trials. If a coin lands heads \(56\) times in \(100\) flips, the experimental probability of heads is \(0.56\).
Practice
Quick check
If \(P(A) = 0.27\), what is \(P(\text{not } A)\)?
\(1 - 0.27 = 0.73\).
Quick check
A spinner lands on blue \(18\) times in \(60\) spins. What is the experimental probability of blue?
\(\frac{18}{60} = 0.3\).
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