Identifying Range of Probability and Complementary Events

By the end of this lesson, learners should be able to

1. Identify that the probability of any event lies in the range 0 ≤ P(N) ≤ 1.
2. Distinguish between impossible events (P = 0) and certain events (P = 1).
3. Calculate the probability of complementary events using P(N') = 1 − P(N).

Key vocabulary

Impossible event

An outcome that can never occur, assigned a numerical probability value of exactly zero.

Getting a seven when rolling a standard six-sided die is an impossible event with $P=0$.

Certain event

An outcome that is guaranteed to happen, having a numerical probability value of exactly one.

Obtaining a number less than seven on a standard die is a certain event with $P=1$.

Complementary event

The event that an outcome does not occur, whose probability equals one minus the original.

If $P(\text{rain})=0.65$, its complementary event has probability $0.35$.

Lesson preview

Slide 1 · The Range of ProbabilityOpening · 18 min

The probability $P(N)$ of any event $N$ measures how likely the outcome is to occur. Probability is always a real number ranging from $0$ to $1$ inclusive, written as $0\le P(N)\le 1$. An event that cannot happen is an impossible event with $P(N)=0$. An event that must happen is a certain event with $P(N)=1$. When we list all distinct possible outcomes of an experiment, the sum of their individual experimental or theoretical probabilities is always $1$. Use Figure 1 to see how probability is measured on a number line from zero to one.

• The probability of any event $N$ satisfies the inequality bounds $0\le P(N)\le 1$ inclusive.
• An impossible event has a probability equal to $0$ and can never occur in any trial.
• A certain event has a probability equal to $1$ and is guaranteed to occur every time.
• The sum of the probabilities of all distinct possible outcomes in an experiment equals $1$.

Worked example. Calculate $P(\text{Heads})=\frac{9}{20}=0.45$. Calculate $P(\text{Tails})=\frac{11}{20}=0.55$. Verify the sum: $P(\text{Heads})+P(\text{Tails})=0.45+0.55=1.00$. Confirm both $0.45$ and $0.55$ lie within $0\le P(N)\le 1$. Answer: $P(\text{Heads})=0.45$, $P(\text{Tails})=0.55$, and their sum is $1.00$..

Check question: A box contains red, blue, and yellow beads. The probability of choosing a red bead is 0.2 and blue is 0.7. What is the probability of choosing a green bead, and what is P(yellow)?

Show answer

P(green) = 0 and P(yellow) = 0.1. Green beads do not exist in the box, so picking green is impossible ($P=0$). Since total probability is $1$, $P(\text{yellow})=1-(0.2+0.7)=0.1$.

Slide 2 · Calculating Probabilities of Complementary EventsPractice · 22 min

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• Both slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 11 lessons in this sub-strand

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All 12 lessons in Probability 1

1. 1Performing Probability Experiments and Finding Experimental ProbabilityPerform experiments involving probabilities in different situations
2. 2Empirical Probabilities and Experimental InvestigationsPerform experiments involving probabilities in different situations
3. 3Identifying Range of Probability and Complementary EventsIdentify the range of probability in different situations
4. 4Generating Probability Space of EventsGenerate probability space of different events
5. 5Generating Probability Space for Compound EventsGenerate probability space of different events
6. 6Determining Probability of Mutually Exclusive Events and Introduction to Independent EventsDetermine the probability of mutually exclusive events and independent events
7. 7Determining Probability of Independent Events and Multi-Stage SelectionsDetermine the probability of mutually exclusive events and independent events
8. 8Probability Bounds, Complementary Events, and the Addition LawApply the laws of probability in different situations
9. 9Applying Laws of ProbabilityApply the laws of probability in different situations
10. 10Determining Probability of Independent Events Using Tree DiagramsDetermine the probability of independent events using tree diagrams
11. 11Tree Diagrams for Multi-Stage Independent EventsDetermine the probability of independent events using tree diagrams
12. 12Applications of Probability in Real-Life ContextsAppreciate the application of probability in real-life situations