Chapter 4: 4.2 - Additional Rules for Probability: Fundamental Properties and the Complement Rule

Fundamental Properties of Probability

• The probability of any event $E$ is bounded such that it is always between $0$ and $1$ inclusive (representing a range from $0\%$ to $100\%$):

  • Inequality notation: $0 \le P(E) \le 1$

• Impossible Event ($P(E) = 0$):

  • Definition: An event that has zero chance of occurring under any circumstance.

  • Symbolic notation: $E = \emptyset$ (the empty set), yielding $P(E) = 0$.

  • Concrete example: Rolling a $10$ on a standard 6-sided die. Because the face $10$ does not exist on the die, the event set is empty ($E = \emptyset$), resulting in a probability of $P(E) = 0$.

• Certain Event ($P(E) = 1$):

  • Definition: An event that is absolutely guaranteed to occur every single time the random experiment is performed.

  • Theoretical condition: A certain event occurs when the event space equals the entire sample space ($E = S$).

  • Formula derivation: $P(S) = \frac{n(S)}{n(S)} = 1$

  • Concrete example: Rolling a number strictly less than $7$ on a standard 6-sided die. The outcomes meeting this condition comprise the entire set of possible rolls: $E = \{1, 2, 3, 4, 5, 6\}$. Since $n(E) = 6$ and $n(S) = 6$, the probability is $P(E) = \frac{6}{6} = 1$.

The Complement Rule ("Not" Probabilities)

• Definition of a Complement:

  • The complement of an event consists of all outcomes in the sample space $S$ that are not included in event $E$.

  • Notation: $E^c$ denotes "not $E$" or the complement of $E$.

• Core Complement Formula:

  • An event and its complement together encompass all possibilities in the entire sample space without overlap.

  • Fundamental identity: $P(E) + P(E^c) = 1$

  • Derived identities:

  • Probability of the complement: $P(E^c) = 1 - P(E)$

  • Probability of the original event: $P(E) = 1 - P(E^c)$

• Comprehensive Example with a Standard Six-Sided Die:

  • Sample Space ($S$): Rolling a single 6-sided die yields $S = \{1, 2, 3, 4, 5, 6\}$.

  • Event ($E$): Rolling an even number corresponds to the subset $E = \{2, 4, 6\}$.

  • Complement ($E^c$): Not rolling an even number corresponds to rolling an odd number, yielding the subset $E^c = \{1, 3, 5\}$.

  • Probability Calculation:

  • The number of favorable outcomes in the complement is $n(E^c) = 3$.

  • The probability is calculated as $P(E^c) = \frac{3}{6} = \frac{1}{2}$.

  • Verification:

  • Combining the outcomes of event $E$ and complement $E^c$ produces the set union: $\{2, 4, 6\} \cup \{1, 3, 5\} = \{1, 2, 3, 4, 5, 6\}$.

  • This combination accounts for every element in sample space $S$, confirming that $P(E) + P(E^c) = \frac{3}{6} + \frac{3}{6} = 1$.