Keywords

Key Questions

Probability of an event A: P(A) = number of favorable cases / number of possible cases.
Probability of the complementary event not-A: P(not-A) = 1 - P(A).

Complementary Events:
- Definition: Two events are complementary when the occurrence of one implies the non-occurrence of the other.
- Example: When rolling a die, if event A is 'getting an even number', the event not-A (complementary) is 'not getting an even number' (i.e., getting an odd number).
Certain and Impossible Events:
- Definition: An event is certain when its probability of occurring is 1, and it is impossible when its probability is 0.
- Example: In a coin toss, the certain event is 'getting heads or tails' and the impossible event would be 'landing on the edge' (considering a regular coin).
Sample Space:
- Definition: The set of all possible outcomes of a random experiment.
- Example: In rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.
Sum of Probabilities:
- Principle: The sum of the probabilities of all possible events within a sample space is always equal to 1 (100%).

Probability of Event A:
- Formula: P(A) = number of favorable cases for event A / total number of cases in the sample space.
- Strategy: Identify the number of outcomes that satisfy event A.
Probability of the Complementary Event (not-A):
- Formula: P(not-A) = 1 - P(A).
- Strategy: Calculate the probability of A and subtract from 1 to find the complementary probability.

Flipping three coins in a row:
- Sample space: Each coin can land heads (H) or tails (T), resulting in 8 possible combinations (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT).
- Event A: 'Getting at least one head'.
- Event not-A (complementary): 'Not getting any heads' (i.e., getting TTT).
- Calculation: P(A) would be the probability of getting at least one head and P(not-A) would be 1/8 (only one of the eight possible outcomes is TTT).
- Using the complementarity rule: To find P(A), we calculate P(not-A) and subtract from 1. Thus, P(A) = 1 - P(not-A) = 1 - 1/8 = 7/8.
Guided Exercise:
- Imagine you have a deck of 52 cards and event A is 'drawing an ace'.
- Sample space: 52 possible cards.
- Favorable cases for A: 4 aces in the deck.
- Calculation: P(A) = 4/52. To find the event not-A ('not drawing an ace'), we calculate P(not-A) = 1 - P(A) = 1 - 4/52 = 48/52.

Summary of Key Points:
- Complementary events are pairs of events where the occurrence of one excludes the occurrence of the other; the sum of their probabilities totals 1.
- Calculating the probability of a complementary event is done by the difference between 1 and the probability of the opposite event.
- Understanding the sample space is crucial as it represents all possible outcomes of a random experiment.
- Applying the formula for the probability of complementary events simplifies the calculation of less obvious or more laborious events to quantify directly.
Conclusions:
- The complementarity rule (P(A) + P(not-A) = 1) is a powerful tool for probability calculations.
- Analyzing complementary events is an efficient approach to dealing with complex probability problems.
- Reasoning based on complementary events allows for a deeper understanding of the random behavior of experiments and real-world situations.
- The ability to calculate probabilities of complementary events and recognize the total sum of probabilities as 1 is fundamental in various mathematical and everyday applications.