Grade 10 · Core Mathematics · Lesson 8 of 12

Probability Bounds, Complementary Events, and the Addition Law

A Grade 10 Core Mathematics CBE lesson from the sub-strand Probability 1. In it, learners state and apply the probability bounds 0 ≤ P(N) ≤ 1 for impossible, possible, and certain events.

Strand Statistics and ProbabilitySub-strand Probability 1Lesson code 10CM-32-L840 minutes · 4 slides

KICD learning outcome (e)

Apply the laws of probability in different situations.

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

State and apply the probability bounds 0 ≤ P(N) ≤ 1 for impossible, possible, and certain events.
Use the complement rule P(N') = 1 − P(N) to solve probability problems in real-life contexts.
Identify mutually exclusive events and apply the addition law P(A or B) = P(A) + P(B) to compound outcomes.

Key vocabulary

Complementary events: Two events that are mutually exclusive and together contain all possible outcomes.; Germinating and not germinating are complementary events whose probabilities add up to 1.
Impossible event: An event that has zero chance of occurring in any trial.; Rolling a 7 on a standard six-sided die is an impossible event.
Mutually exclusive events: Events that cannot happen simultaneously in a single trial or outcome.; Rolling an even number and rolling a 5 are mutually exclusive events.
Addition law: The rule stating $P (A or B) = P (A) + P (B)$ for mutually exclusive events.; Using the addition law, $P (orange or lemon) = P (orange) + P (lemon)$ .

Lesson preview

Slide 1 · Probability Bounds and Complementary EventsOpening · 10 min

Figure 1: A probability scale from 0 to 1 and a Venn diagram illustrating complementary events that add up to one.

The probability of any event $N$ is a number from 0 to 1 inclusive, written $0 \leq P (N) \leq 1$ . If an event cannot happen, $P (N) = 0$ , making it an impossible event. If an event is guaranteed to occur, $P (N) = 1$ , making it a certain event. Since an event either occurs or does not occur, the event $N$ and its complement $N^{'}$ represent mutually exclusive and exhaustive outcomes. Therefore, $P (N) + P (N^{'}) = 1$ or $P (N^{'}) = 1 - P (N)$ .

An event $E$ 's probability lies between 0 and 1 inclusive, written $0 \leq P (E) \leq 1$ .
An event with probability 0 is impossible, while an event with probability 1 is certain.
The complement $N^{'}$ represents the event that $N$ does not occur in a trial.
Because $N$ and $N^{'}$ cover all outcomes, $P (N) + P (N^{'}) = 1$ and $P (N^{'}) = 1 - P (N)$ .

Worked example. Let

G

be the event that the bean seed germinates, so

P (G) = \frac{7}{8}

. The complement

G^{'}

is the event that the seed does not germinate. Use the complement law:

P (G^{'}) = 1 - P (G) = 1 - \frac{7}{8} = \frac{1}{8}

. By definition, an outcome with probability 0 is an impossible event. Answer: The probability of not germinating is $\frac{1}{8}$ , and a probability 0 outcome is impossible..

Check question: If the probability that it rains at a weather station on Mashujaa Day is 0.35, what is the probability that it does not rain?

0.35
0.55
0.65
0.75

Show answer

0.65. Using the complement law, $P (no rain) = 1 - P (rain) = 1 - 0.35 = 0.65$ .

Slide 2 · Applying Complement Rules in Practical SituationsModel · 10 min

Figure 2: Two examples of complementary events, showing the probabilities of not rolling a six on a die and picking green marbles.

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All 4 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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Probability Bounds, Complementary Events, and the Addition Law

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 12 lessons in Probability 1