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Unit 4: Data, Statistics, and Probability

Topic 4

Basic and Compound Probability

Probability measures how likely an event is to occur:
P(event)=number of favorable outcomestotal possible outcomes.P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}.
Probabilities range from 00 (impossible) to 11 (certain). For independent events, probabilities multiply. For mutually exclusive events, probabilities add.
Basic Rules
  • P(A′)=1−P(A)P(A') = 1 - P(A)
  • P(A or B)=P(A)+P(B)−P(A and B)P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and }B)
  • P(A and B)=P(A)⋅P(B)P(A\text{ and }B) = P(A)\cdot P(B) if AA and BB are independent

Core Skills

  • Compute simple probabilities for dice, cards, and spinners.
  • Distinguish between “or” (addition) and “and” (multiplication).
  • Identify whether events are independent or dependent.
  • Use complementary probability for “at least one” problems.

Example 1: Basic Probability

A fair die is rolled. P(even)=36=12P(\text{even}) = \dfrac{3}{6} = \boxed{\tfrac{1}{2}}.

Example 2: Independent Events

Flip a coin and roll a die. P(heads and 4)=12×16=112P(\text{heads and 4}) = \tfrac{1}{2}\times\tfrac{1}{6}=\boxed{\tfrac{1}{12}}.

Example 3: Dependent Events

A box has 3 red and 2 blue balls. Draw 2 without replacement. P(both red)=35×24=310P(\text{both red})=\tfrac{3}{5}\times\tfrac{2}{4}=\boxed{\tfrac{3}{10}}.

Example 4: “At Least One”

Two coin flips. P(at least one head)=1−P(no heads)=1−(12)2=34P(\text{at least one head})=1-P(\text{no heads})=1-(\tfrac{1}{2})^2=\boxed{\tfrac{3}{4}}.

Example 5: “Or” Probabilities

In a standard deck, P(heart or king)=1352+452−152=1652=413P(\text{heart or king})=\tfrac{13}{52}+\tfrac{4}{52}-\tfrac{1}{52}=\boxed{\tfrac{16}{52}=\tfrac{4}{13}}.

Example 6: Compound Probability (Tree Diagram Idea)

A coin is flipped twice. The possible outcomes are HH, HT, TH, TT. P(exactly one head)=24=12P(\text{exactly one head})=\tfrac{2}{4}=\boxed{\tfrac{1}{2}}.

Key Takeaways

  • Add probabilities for “or,” multiply for “and” (independent).
  • Use complements for “at least” or “none” types.
  • For dependent events, adjust denominators as items are removed.
  • All probabilities must sum to 1 for the entire sample space.