🎲 Probability

How likely is it? Measure the chance!

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What is Probability?

Probability tells us how likely something is to happen. It is always a number from 0 (impossible) to 1 (certain). We can write it as a fraction, a decimal, or a percent.

\(P(\text{event}) = \dfrac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\)
0
Impossible ¼
Unlikely ½
Even chance ¾
Likely 1
Certain

A Worked Example

A bag has 3 red and 2 blue marbles. What is the probability of drawing a red one?
Favorable outcomes (red) = 3
Total outcomes = 3 + 2 = 5
\(P(\text{red}) = \dfrac{3}{5} = 0.6 = 60\%\)

Key Ideas

Complement (NOT)

\(P(\text{not } A) = 1 - P(A)\)

If \(P(\text{rain}) = 0.3\), then \(P(\text{no rain}) = 0.7\).

Independent events (AND)

\(P(A \text{ and } B) = P(A) \times P(B)\)

Two coins both heads: \(\tfrac12 \times \tfrac12 = \tfrac14\).

Mutually exclusive (OR)

\(P(A \text{ or } B) = P(A) + P(B)\)

Roll a 1 or a 2: \(\tfrac16 + \tfrac16 = \tfrac26 = \tfrac13\).

Theoretical vs Experimental

Predicted vs actually observed

The more trials you run, the closer experimental gets to theoretical.

Common Outcome Counts

🪙 Coin

2 outcomes: Heads, Tails. \(P(\text{H}) = \tfrac12\).

🎲 Die

6 outcomes: 1–6. \(P(4) = \tfrac16\).

🃏 Card deck

52 cards. \(P(\heartsuit) = \tfrac{13}{52} = \tfrac14\).

Tips

  • Probability is never less than 0 or more than 1.
  • All the probabilities of every outcome add up to 1.
  • Always simplify the fraction (e.g. \(\tfrac{2}{6} = \tfrac13\)).
  • "Favorable" just means the outcome you are asking about — not always a "good" one.