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Explore basic concepts of probability, including calculating the likelihood of events and understanding sample spaces.

What Are the Chances: Probability

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain), or equivalently as a percentage from 0% to 100%. From weather forecasts and medical diagnoses to games and insurance, probability provides the mathematical framework for reasoning about uncertainty and making informed decisions.

Components of Probability

This section covers the foundational concepts:

  • Basic Probability: P(event) = number of favorable outcomes / total number of outcomes. A fair coin has P(heads) = 1/2.
  • Compound Events: The probability of two events both happening (AND) or at least one happening (OR), using multiplication and addition rules.
  • Independent vs Dependent Events: Independent events do not affect each other's probability; dependent events do (the second probability changes after the first occurs).
  • Complementary Events: The probability of an event NOT happening is 1 minus the probability of it happening: P(not A) = 1 - P(A).

Examples of Probability

Basic Probability Examples

  • A standard die has 6 faces. P(rolling a 4) = 1/6 β‰ˆ 0.167 or about 16.7%.
  • A bag has 3 red and 7 blue marbles. P(red) = 3/10 = 0.3 or 30%.
  • A deck of 52 cards has 4 aces. P(drawing an ace) = 4/52 = 1/13 β‰ˆ 7.7%.

Compound Event Examples

  • P(rolling a 3 AND then a 5 on two dice rolls) = 1/6 Γ— 1/6 = 1/36 β‰ˆ 2.8% (independent events, multiply).
  • P(drawing a heart OR a king from a deck) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13 β‰ˆ 30.8% (subtract the king of hearts counted twice).
  • A spinner has 3 equal sections: red, blue, green. P(red OR blue) = 1/3 + 1/3 = 2/3 β‰ˆ 66.7%.

Independent vs Dependent Examples

  • Flipping a coin twice: P(heads then heads) = 1/2 Γ— 1/2 = 1/4. The first flip does not affect the second (independent).
  • Drawing two cards without replacement: P(ace then ace) = 4/52 Γ— 3/51 = 12/2,652 = 1/221 β‰ˆ 0.45% (dependent β€” fewer aces and cards remain).
  • A bag has 5 red and 5 blue marbles. Drawing red first without replacement changes P(red on second draw) from 5/10 to 4/9 (dependent).

Complementary Event Examples

  • P(not rolling a 6) = 1 - 1/6 = 5/6 β‰ˆ 83.3%.
  • If the probability of rain tomorrow is 0.35, the probability of no rain is 1 - 0.35 = 0.65 or 65%.
  • A student has a 92% chance of passing an exam. The chance of not passing is 1 - 0.92 = 0.08 or 8%.